Collected Scientific Papers

S N Bose : The Man and His Work

Part 1 : Collected Scientific Papers


Part I : Collected Scientific Papers

S N Bose National Centre for Basic Sciences Calcutta

EDITORS C K Majumdar Partha Ghose

Enakshi Chatteqjee Samik Bandyopadhyay

Santimay Chatterjee (Principal Editor)

Q S N Bose National Centre for Basic Sciences, Calcutta 1994

Qpeset and printed by Graf-M Publisher, 3 Ashu Biswas Road, Calcutta 25, India

Contents

Acknowledgements

Foreword

Introductions t o Scientific Papers

1 The Equation of State of a Real Gas C K MAJUMDAR

2 Classical Mechanics CKMAJUMDAR

3 Quantum Theory E C G SUDARSHAEl

4 Chemistry MIHIR CHOUDHURY B R NAG

6 Statistics S P MWKHERJEE

7 The Ionosphere EDITORS

8 Mathematical Physics N MUgUNDA

9 Unified Field Theory A K RAYCHAUDHURI

10 Thermoluminescence H N BOSE

11 Bose Statistics : a historical perspective PARTHA GHOSE

Collected Scientific Papers

1 On the Influence of the Finite Volume of Molecules on the. Equation of State

2 The Stress-Equations of Equilibrium

3 On the Herpolhode

4 On the Equation of State

5 On the Deduction of Rydberg's Law from the Quantum Theory of Spectral Emission

6 Plancks Gesetz und Lichtquantenhypothese

7 Planck's Law and the Light-Quantum Hypothesis

8 Wiirmegleichgewicht im Strahlungsfeld bei Anwesenheit von Materie

9 Thermal Equilibrium in the Radiation Field in the Presence of Matter

10 Messungen der Zersetzungsspannung in nichtwasserigen L(lsungsmitte1n

11 Measurement of the Decomposition Voltage in Nonaqueous Solvents

12 Beryllium Spectrum in the Region h 3367-1964

13 Tendencies in the Modern Theoretical Physics

14 On the Complete Moment-coefficients of the D~-statistic

15 On the Moment-coefficients of the D~-statistic and Certain Integral and Differential Equations Connected with the Multivariate Normal Population

16 Recent Progress in Nuclear Physics

17 Anomalous Dielectric Constant of Artificial Ionosphere

18 On the Total Reflection of Electromagnetic Waves in the Ionosphere

19 Studies in Lorentz Group

20 The Complete Solution of the Equation :

21 Reaction of Sulphonazides with Pyridine : Salts and Derivatives of Pyridine-Imine

22 A Note on Dirac Equations and the Zeeman Effect

23 The Classical ~eterminism and the Quantum Theory

24 On an Integral Equation Associated with the Equation for Hydrogen Atom

25 Germanium from Sphalerite

26 Extraction of Germanium from Sphalerite Collected from Nepal- Part I

27 Extraction of Germanium from Sphalerite Collected from Nepal- Part I1

28 Les Identius de Divergence dans la Nouvelle Theorie Unitaire

29 The Identities of Divergence in the New Unified Theory

30 Une Th6orie du Champ Unitaire avec Tp t 0

31 A Unitary Field Theory with rp 'lr 0

32 Certaines Cons6quences 1' Existence du Tenseur g dans le Champ Affine Relativiste

33 Certain Consequences of the Existence of the Tensor g in the Affine Relativistic Field

34 The Amne Connection in Einstein's New Unitary Field Theory

36 A Report on the Study of Thermoluminescence

36 Solution d'une $quation Tensorielle Intervenant dam la Th6orie du Champ Unitaire

37 Solution of a Tensor Equation Appearing in the Unitary Field Theory

Acknowledgements

INSTITUTIONS

The Asiatic Society, Calcutta Indian Association for the Cultivation of Science, Calcutta Indian National Science Academy, New Delhi Indian Statistical Institute, Calcutta Bangiya Vijnan Parishad, Calcutta Bibliotheque Nationale, Paris Council of Scientific and Industrial Research, New Delhi Dhaka University, Dhaka The Jewish National and University Library, Jerusalem Musee Curie, Institut Curie, Paris National Physical Laboratory, New Delhi National Library, Calcutta Nehru Memorial Museum and Library, New Delhi Rabindra Bhavana, Visva-Bharati, Santiniketan Rajya Sabha Secretariat, New Delhi The Royal Society, London Saha Institube of Nuclear Physics Library, Calcutta Variable Bnergy Cyeotron Centre Library, Calcutta -for giving us access to information, material and documents in their holdings

THE FAMILY OF S N BOSE

and in particular Rathindranath and Benu Bose, Nilima Mitra, Ruchira Mitra, Ramendranath Bose, and Jayanti Chattopadhyay

PUBLISHERS AND EDITORS OF JOURNALS

that originally carried S N Bose's papers appearing in this volume

AUTHORS

of the Introductions to the Collected Scientific Papers

TRANSLATORS

S V Raman, Biswarup Banerjee, Sangeeta Mitra for translations of papers and correspondence in German Agnes Rozario and M Schillings, sj for translations of papers i n French Somjit Datta for translations of articles in Bengali

PHOTOGRAPHERS

Nemai Ghosh, George Ponodath sj, Chitrabani

OTHER INDMDUALS

Debajyoti Datta, Tandra Datta, Kalyani Ghose Emazuddin Ahmed, Vice-chancellor, Dhaka University, Abu Zaid Sikdar, Registrar, Dhaka University,

K M Mannan, Chairman, Physics Department, Dhaka University qjay Ray, Lalit Nath and other members of the Physics Faculty, Dhaka University, Quazi A K M Mainuddin, photocopier operator a t the Registrar's Office, Dhaka University Dilipkumar De, Hena Basu, Pronoti De, Mrinalkumar Dasgupta, Arunkumar Dasgupta, Sarojranjan Chakravarty, Sadhan Dasgupta, Ranatosh Chakravarty, Amitabha Chakravarty, Jyotirmoy Gupta, Dilip Malakar, Sandhya Mitra, Jugalkanti Ray, Jayanta Basu Ram Singh 'Ibmar, Jyotirmoy Sanyal, Somendranath Bandyopadhyay and Dipankar Chattarji, all of Visva-Bharati Patrick Petitjean i n Paris Sanjida Khatun and Abdur Rajjak i n Dhaka Abhijit Datta and Prabhat Thakur Hiralal, Professor Bose's old bearer

S N Bose : Introductions to

scientific papers

Foreword

The Satyendra Nath Bose National Centre for Basic Sciences was established by the Department of Science and Technology, Government of India, in 1986 as a memorial to the Indian scientist Satyendranath Bose. On the occasion of his birth centenary in 1994, the Centre has prepared the present volume incorporating his scientific papers, some selected miscellaneous pieces and addresses, and a biography. Bose wrote and published in four languages : Bengali, English, German and French. Though we have included the original German and French articles, we have provided their English translations also. His writings in Bengali have been collected and brought out by Bangiya Vijnan Parishad, an institution he helped to establish. We have included translations of some of these articles which may be of interest to a general reader.

Several biographical accounts of S N Bose were available. We found that the list of scientific papers was incomplete in all of them, and that not enough attention had been given to his work in Dhaka after his return from Europe. Two published papers -one in German (1927) and another in English (1929) from this period - are included in our collection. Enakshi Chatterjee and Santimay Chatterjee have given a fuller account of his activities in Dhaka - establishing physics laboratories, and helping develop modern research in physics and chemistry.

In some respects S N Bose's career is the story of the triumph and tragedy of Indian science iii the first half of the twentieth century. His brief encounter with Europe in the twenties assured his place among the rea t scientists. He tried t o strengthen the scientific base in India but he found little financial and spiritual sustenance. He himself described these two aspects very well, when he said that he was like a comet which came once but never returned, and he seemed to be living on the Moon.

Some comments by later Indian workers about the scientific papers are also included; these hopefully will help an inquisitive young research worker in placing the

2 .S N Bose : The Man and His Work

scientific papers in the right perspective. A long list of acknowledgements preceding this foreword covers most of the indi-

viduals and organizations who have helped us in several ways. Special thanks are due t o the Department of Science and Technology, Government of India, for financial support to the S N Bose National Centre for Basic Sciences for activities in the Centenary year, of which the present publication is one.

The volume is divided into two parts - the first part aimed a t a more specialist readership, containing his scientific papers, with introductory commentaries by later workers in the specific .fields; and the second part designed for a more general readership, with the exhaustive biography, his public lectures and addresses, and miscellaneous pieces (several of them translated from Bengali for the first time).

C K MAJUMDAR Director

S N Bose Natioaal Centre for Basic Sciences Calcutta

The Equation of State of a Real Gas

1 On the Influence of the Finite Volume of Molecules on the Equation of State (1918) 2 On the Equation of State (1920)

In a doctoral dissertation to the University of Leiden in 1873, 3 D van der Wads proposed a modified equation of state of a real gas. He included two effects : the excluded volume effect due to strong range repulsion and an intrinsic pressure due to weak long range attraction. The work was praised by J C Maxwell who proposed the well-known Maxwell construction to remove the unstable part of the isotherms. The van der Waals equation was enormously successful in explaining critical phenomena, critical indices and the liquefaction of gases. By 1910 experimentalists, however, began to find deviations from the predictions; two constants were not enough, and people started looking for improvements. In 1901 K Onnes introduced the virial coefficients to represent the isotherms. But the simplicity of the approach of an equation of state with a few parameters was too attractive : more than fifty equations of state have been proposed. M N Saha and S N Bose used thermodynamics, especially the Boltzmann formula for entropy, in their characteristic way to arrive a t their equation of state. The Boltzmann formula would appear again in Bose's other works. The modern theory of a gas a t high density starts from the cluster development of H D Ursell and J E Mayer (J E Mayer and M G Mayer, Statistical Mechanics, J Wiley and Sons, New York 1977, Second edition, pp.229 ff.).

C K MAJUMDAR S N Bose National Centre for Basic Sciences, Calcutta.

Classical Mechanics

1 The Stress-Eqwtions of Equilibrium (1919) 2 OntAcHerpolhodc(1919)

The equilibrium of isotropic elastic solid bodies is discussed in the well-known book by AE H Love,A Deatise On the Mathematical Theory ofElasticity, Fourth Edition, Dover, New York 1927, ch V Here in pp. 134-5, J H Michell's equations are given. V Cerutti's method is described in ch X, pp. 237-40. The problem of the spherea is discussed in ch XI. After the publication of the paper in 1919 Bose lost interest in the problems of elasticity and never returned to them in published work.

The force-free motion of a body about a fixed point is discussed in books of classical dynamics (eg E T Whittaker, A Deatise on the Dynamics of Particles and Rigid Bodies, Fourth edition, Cambridge University Press, Cambridge 1961) in terms of elliptic functions. Poinsot's construction gives a simple geometrical description of the motion. Bose proved a geometrical property without using elliptic functions. The same method was also found by W van der Woude.

C K MAJUMDAR S N Bose National Centre for Basic.Sciences, Calcutta.

Quantum Theory

1 On the Deduction of Rydberg's Law from the Quantum Theory of Spectral Emission (1920)

2 Plancks Gesetz und Lichtquantenhypothese (1924) 3 WLirmegleichgewicht im Strahlungsfeld bei Anwesenheit von Materie (1924) 4 A Note on Dirac Equations and the Zeeman Effect (1943)

Bose's contribution to Quantum Theory consists of four papers, two of which are on atomic physics and two are about quantum statistical mechanics. This covers the period 1920-1943. Of these, the best known is the paper on Bose statistics for the photon gas (2 f Physik 26, 178 (1924)). A brief critical appreciation of these papers follows:

In the f ~ s t paper On the Deduction of Rydberg's Law from the Quantum Theory of Spectral Emission, the Bohr-Sommerfeld method of quantization of action

is used to generate Balmer terms

for any series for any atom. Bose starts with a monopole plus a dipole as the best approximation of the potential seen by a valence electron :

8 N Bose : The Man and His Work

e2 e L cos 0 V ( r ) = --+

r r2

The calculations involve abelian integrals which have to be suitably%pproximated. Bose also has an effective expression for the Rydberg constant in terms of the fundamental constants, but is either unaware of (or disinclined to cite) A Haas who first obtained the relation ( A Haas, quoted in Introduction to Theoretical Physics, vol 11, Constable, London 1948).

In the paper A Note on Dirac Equations and the Zeeman Effect (Indian J Phys. 17, 301 (1943)), written together with K Basu, the problem of the energy levels of a hydrogenic atom in an inhomogeneous magnetic field is solved using Sonine polynomials. The calculations are elegant and straightforward and lead to the quadratic dependence of the perturbed energy levels on the magnetic field.

The celebrated paper Plancks Gesetz und Lichtquantenhypothese ( Z f Physik 26,168 (1924)) of S N Bose introduces the new photon statistics and deduces the Planck distribution as the equilibrium distribution for photons. Bose considered complexions of photons as the primary characterization of the state rather than which photon had which energy. Photons were treated as strictly indistinguishable.

With the modification that Bose introduced into the calculation we obtain the correct thermodynamics of the photon gas. That one step was the basis of the new synthesis between the wave and the particle properties of photons, and with it, the foundations of quantum field theory. After Bose's paper came an avalanche of developments: the extension of Bose's theory to particles of arbitrary mass and non-zero chemical poten. tial by Einstein, the Fermi-Dirac statistics for electrons, the quantization of the electromagnetic field by Heisenberg and Pauli, and quantum electrodynamics by Dirac.

To put Bose's synthesis in its proper setting it is good to recall that in the nineteen- twenties there were two items of unfinished business, one regarding photons as particles and the other concerning statistical mechanics of identical-particles. In both cases uneasy makeshift solutions were.generally accepted instead of definitive solutions. In the old dichotomy between particles and waves for describing light there were highly persuasive arguments on both sides, but i t was thought that a crucial experiment was the determination of the relative speed of light in two media with different refractive indices. I t appeared that if light consisted of particles, the speed in the optically denser medium should be greater; if i t consisted of waves,this speed should be lesser. Fizeau's experiment on the speed of light in water thus seemed to find evidence definitively for the wave theory! Yet, what of the postulated photons? How do we reconcile the notion of a photon with its discrete momentum hnd energy to its lesser speed in water than in air? We must conclude that photons may be particles, but they do not behave as particles are naively expected to behave. A revision of the concept of a particle ought to be made. We now know that extended particles also bend towards the normal in a medium in which i t moves slower; so Fizeau's experiment could not really distinguish between extended particles and waves.

Quantum Theory 9

The other item of unfinished business is even older. I t concerns itself with the statistical mechanics of identical particles. In calculating the partition function and the entropy, one finds that the entropy is not a strictly additive qua&ity: when we mix two volumes of an ideal gas a t the same temperature and pressure the resultant entropy is larger than the sum of the two entropies.This Gibbs' paradox shows that such a collection of identical particles is not a satisfactory model for an ideal gas. Instead of heeding this warning signal people 'fixed' the trouble by an ad hoc procedure in dividing the partition function by the factorial of the number of particles, thus condoning the Maxwell--Boltzmann statistics. The genuine need for a reexamination of the implications of the strict identity of the particles was not appreciated until Bose, three decades yet to be born.

Elementary particles were originally introduced as the stuff from which the world was made. They were to be immutable entities. But the photon was clearly an entity which could be created or destroyed. Where does a photon come from and where does it go? And how can we really understand creation and destruction? What is the implication of strict identity of photons? In what sense and to what extent can we think of light as a collection of photons? All these questions were answered a t one stroke by Bose, who asked us to consider the many-photon states to be counted as states with equal probability. Photons were thus particles all right, but particles for which strict identity was to be recognized by considering as distinct only those cases in which the distribution of photons over phase cells were distinct.

Photons thus became nothing but levels of an underlying field. Creation or destruction of photons is merely a 'movement' of the field. Photons are then manifestations of the potentialities of the radiation field: the dichotomy between the field and the particle thus ceases. Two have become one.

Automatically the embarrassment of the Gibbs paradox is resolved: the paradox was just telling us that the strict identity of particles must be taken into account. But if photons are but the differences between the levels of the radiation field, they are all identical! And the process of creation and destruction is thought of as a change in the state of the field, the 'motion' of the field. If we have equations of motion of the field, we have the means of describing the creation and destruction of photons. I t took two more years for Heisenberg and Pauli to write down the equations of motion for the radiation field and another year for Dirac to construct a theory of the emission and absorption of photons. In Dirac's work the oscillators of Planck were at last identified. The formulation of the equations of motion of the electromagnetic field had still unsatisfactory features. Many others contributed to the resolution of this problem, among them Dirac, Heisenberg and Pauli, Fermi and Gupta.

In the course of his work on the quantum theory of radiation Dirac introduced the now familiar notion of creation and destruction operators which increase or decrease the number of quanta in a state. These creation and destruction operators, introduced as the operator coefficients of the quantized field operator, do not commute with each other but instead satisfy a commutation relation which transcribes the commutation relations between field quantities as formulated by Heisenberg and Pauli. Dirac had

10 S N Bose : The Man and His Work

already discovered that the commutator bracket in quantum mechanics was the natural analogue to the Poisson bracket in classical mechanics. I t reaffirms the Bose hypothesis that photons obey the Bose statistics.

When Bose advanced his hypothesis the only species of elementary particles that were identified were the electron, the proton, the neutron and the photon. Of these only the photon obeys Bose statistics. The photon number is not covered and i t is a zero mass particle. Both of these impart special characteristics to the statistics of photons. Among the Bose systems available then was Helium. In this case the particle number is conserved; and the particles are nonrelativistic. So we need to extend Bose's ansatz. This was done by Einstein in the same year. To conserve the particle number we have to introduce a non-zero chemical potential. For photons the chemical potential is zero. In the case of an ideal Bose gas with non-zero chemical potential there is a critical temperature below which a finite fraction of the gas condenses into a single quantum state. This condensed phase, discovered by London, should exhibit superfluid properties; and London suggested that superfluid helium should be related to this Bose-Ein- stein condensation phenomenon.

Einstein arranged for the translation and publication of Bose's paper on the statistics of photons and added a remark endorsing it as 'substantial progress.' Both in his original letter to Einstein and in his subsequent correspondence Bose addressed the great man as 'teacher' and accords him great respect; and that is as i t ought to be. I t is in the definition of the teacher, as understood in the classical Indian tradition, that he remove all the doubts of the student and weld his understanding into a harmonious unity: such a teacher is the one worthy of adoration.

To that Teacher who removes all my doubts, welds my vision into a unity and thus enables me to gaze on secret knowledge; to that One my homage.

Einstein does not seem to have told Bose how his theory could be extended to a theory of ideal Bose gases by introducing a chemical potential and making use of a general energy-momentum relation. Einstein formulated this extension in one of his papers.

Bose followed up this paper by another more detailed and more ambitious paper. In his first paper, Bose had used a 'static derivation' of an equilibrium configuration as the most probable configuration. In the second paper, he used a 'dynamic derivation' in which the equilibrium configuration is the one in which transitions into and out of each state balance each other. For the special model of a two-level Bohr atom and monochromatic radiation, Einstein had shown (Phys Z 18, 12 (1917)) how one can understand the Planck distribution when one takes into account both the stimulated and the spontaneous emission on the one hand and stimulated absorption on the other.

Bose generalized this to arbitrary atoms with arbitrary numbers of discrete energy levels and radiation of all possible frequencies: the essential elements in the derivation are the conservation of energy in collisions and the ratio of the rates of emissions and absorption. These were correctly computed by Bose. (Unfortunately, for some strange reason, Bose seems to have different values for the absolute transition rates which prompted Einstein to append a critical comment to the paper.)

Quantum Theory 11

In these two papers done before the birth of quantum mechanics as we know today, Bose laid the foundations of a quantum theory of the electromagnetic field.

E C G SUDARSHAN University of Austin at Texas

Chemistry

1 Messungen der Zersetzungsspannung in nichtwiisserigen Ltisungsmitteln (1927) 2 Reaction of Sulphonazides with Pyridine (1943)

To an outsider it might seem odd that a theoretical physicist of Bose's stature, who was intensely trying to understand the basic laws of physics, would now and then take time off and spend days in a dingy chemical laboratory. One might reasonably expect a theoretical physicist like him to get interested in the nature of the chemical bond or in the properties of electrolytic solutions -a problem fruitfully tackled by his friend and classmate J C Ghosh; but strangely, he showed no interest in any of these theoretical challenges. Rather, he liked to synthesize and analyze chemicals useful to contempo- rary society. He was obviously guided by a nationalistic feeling. A part of his interest in down-to-earth chemistry might have been inherited from his father who founded a small chemical industry. His close association in his formative days with Acharya P C Ray, who championed the cause of Indian chemical industries, might also have played a role.

In Dhaka University he set up a working organic chemistry laboratory and encouraged his students to synthesize a number of important drugs like emetine, sulpha drugs, etc.Most of these works are either not published a t all or published without his name. Only occasionally did his name appear, such as in Science and Culture with P K Dutta, where the reaction between sulphonazides and pyridine was studied. His interest in organic chemistry continued after his return to Calcutta University. In an

Chemistry 13

article in Science and Culture 10,1974, p. 295, AChatterjee has recalled some of Bose's forays in organic chemistry, particularly alkaloid chemistry.

At Dhaka, Biswas and Bose developed a simple instrument to measure the polarization voltage and decomposition voltage of some electrolytes in non-aqueous solvents, where a simple manual commutator was used to alternate the direction of current and thus reduce polarization errors a t the electrodes. At Calcutta University he set up an X-ray laboratory and helped chemists with structural problems. Professor J N Muk- herjee, the noted colloid chemist and classmate of Bose, prepared a map of clays of different parts of India. The structural aspects of these clays were studied in Bose's laboratory. He encouraged the organic chemists to determine molecular structures through X-ray analysis.

Although Bose's works in chemistry do not carry the mark of a genius, these show the versatility of the man and his eagerness to solve the problems of the country as well as his colleagues and students.

MIHIR CHOUDHURY Indian Association for the Cultivation of Science,Calcutta

3 Germanium in Sphalerite from Nepal (1950) 4 Extraction of Germanium from Sphalerite Collected from Nepal-Part I (1950) 5 Extraction of Germanium from Sphalerite Collected from Nepal--Part I1 (1950)

The invention of the transistor brought to reality the inventor's dream of realizing an electron device by controlling the motion of electrons available in ample quantity inside a solid. Power was no longer required to be wasted in obtaining controllable electrons in vacuum. The active device for electronics could now be made much smaller and the life expectancy of electronic equipment made much larger.

The first transistor was made with polycrystals of germanium (element number 32 in the Periodic Table) discovered by the German chemist Winkler in 1886 and named aRer his country. As germanium transistors were expected to replace vacuum tubes in all electronic circuits, S K Mitra concluded in his Presidential Address a t the Forty- second Indian Science Congress held in 1955 a t Baroda that 'the future stage, if one may venture to make a prophecy, will be the era of the uses and applications of the element germanium.' But germanium was mostly recovered from chimney dust collected from refineries of lead, titanium and zinc in the USA and in the UK from chimneys of all kinds of industries using coal which in Britain had a large germanium content. There were not many other sources of germanium and so scientists were exploring all possible sources. I t is no wonder that Bose, being aware of all current developments, endowed with a very sharp foresight, and committed to the national interests, encouraged his student R K Datta to look for sources of germanium in minerals. This work resulted in three letters, in which is described tAe procedure used


for extracting germanium from sphalsrite collected from Nepal and the suitability of the ores as a source of germanium. The three letters should be considered as important original contributions to the technology of germanium and establishing sphalerite as a good source of germanium. The only pity is that the work was not apparently followed up, and no semiconductor industry grew in India, although Bose did identify the problem within a year of the invention of the transistor.

B R NAG Institute of Radiophysics and Electronics

University of Calcutta

Spectroscopy

Beryllium Spectrum in the Region h 3367-1964 (1929)

The spectrum of the beryllium atom is similar t o that of the helium atom and should have been easier to analyze. I t is well-known that He I shows singlets and triplets and He I1 (with one electron detached) shows doublets. But early observations did not resolve the triplets clearly. One sees the same kind of controversy in Be I and Be I1 in this paper.

The way the spectra were clarified by earlier workers seems almost miraculous today. We also note that after returning fi-om Europe in 1926, Bose organized a spectroscopic laboratory a t Dhaka. (Another paper published in 1927 deals with experimental work in-electrochemistry.)

C K MAJUMDAR S N Bose National Centre for Basic Sciences, Calcutta

Statistics

1 On the Complete Moment-coefficients of the @-statistic (1936) 2 On the Moment-coefficients of the @-statistic and Certain Integral and Differ-

ential Equations Connected with the Multivariate Normal Population (1937)

Though Statistics -as a scientific method -was known in India even a t the beginning of the present century, significant statistical studies and researches were initiated during the twenties by the late Professor P C Mahalanobis. While working as a Professor of Physics a t Presidency College, Calcutta, Mahalanobis took up several data-analytic studies on a wide variety of real-life problems and followed them up to develop several theoretical models and inferential tools. He could inspire quite a few brilliant young men in physics and mathematics to work on theoretical as well as applicational aspects of Statistics. Some of those drawn to Statistics - directly or indirectly by Professor Mahalanobis - later changed their initial academic pursuits to concentrate on Statistics. Others, like Professor Satyendranath Bose, continued with their original vistas but made remarkable contributions t o statistical theory and absorbed statistical concepts and tools in their subsequent works.

While working on some anthropometric data on statures of Anglo-Indians, Professor Mahalanobis (1936) came up with an idea of generalized distance between two populations -more or less on the lines of Karl Pearson's coefficient of racial likeness (1921). He defined the famous D~-statistic for this purpose in terms of the mean vectors (of the characteristics measured in the two groups) and the covariance matrix. I t was intended t o be a quantity determined entirely in terms of the sample values of the variates. And

Statistics 17

for judging the significance or otherwise of such a sample value of the statistic its sampling distribution had to be worked out. R C Bose (1936) derived an exact distribution of a modified form of the D~-statistic in which population variances and covariances were substituted for the corresponding sample estimates. He used the expression of the density function (in terms of Bessel functions) to obtain the moment coefficient, which turned out to be the same as those calculated earlier by Professor Mahalanobis (1936), using approximate methods.

Fascinated by the new idea of generalized distance and the nuances of the mathematics used by R C Bose, S N Bose (then working in the University of Dacca) looked a t the problem (1936) and came up with a recurrence formula for moments of the modified D2-statistic to obtain the exact moments without using the density expression and investigated some properties of the moment coefficients. He could also offer a different equation which can yield moments of even fractional orders (expressible not by a polynomial but by an infinite series) -entities that have found many important uses later.

Professor Bose continued his interest in the mathematics of this problem. In his 1937 SankhyE aper he started with some algebraic identities among the moments of l' the modified D -statistic deduced fkom their differential forms and derived an integral equation connected with these moments. He investigated the fundamental differential equation satisfied by the multivariate normal distribution in its various forms - spherical, parabolic and product. He also offered a series solution of the differential equation.

The contents of the two papers in Sankhyii (1936 and 1937) speak of the depth of S D o s e ' s vision about a new problem and its mathematical fallouts. I t may be pointed out, however, that the results derived by S N Bose have not created a big impact on statistical aspects of multivariate analysis, since the basic shortcoming of the modified D2-statistic considered by him renders the results somewhat less useful. The differential equation satisfied by the multivariate normal distribution has not found its way into later investigations on characterizations. In fine, the papers by S N Bose on D'-statistic have been rarely cited or used.

S P MUKHERJEE Centenary Professor of Statistics

Calcutta University

References 1 P C Mahalanobis : 'On Tests and Measures of Divergence', Jout: Asiat.Soc.Beng. XXXVI

(1930), 541-588. 2 P C Mahalanobis : 'On the Generalized Distance in Statistics', Proc.Nat.Znst.Sc.India 2

no.1 (1936), 49-55. 3 Rajchandra Bose : 'On the Exact Distribution and Moment- coefficients of the D~-statistic',

Sankhyu 2, Part 2 (1936). 143-154. 4 Rajchandra Bose : 'On the Distribution of Differences of Mean Values in Two Samples and

the Definition of the tatisti is tic', Sankhyu 2, Part 4 (1936), 379-384.

Statistics 18

6 Satyendranath Bose : 'On the Complete Moment-coefficients of the D'-statistic, SankhyE 2, Part 4 (1936), 385-396.

6 Satyendranath Bose : 'On the Moment-coefficients of the D'-statistic and Certain 1nte&l and Differential Equations Connected with the Multivariate Normal Population', Sankhyu 8, Part 2 (1937), 105-124.

The Ionosphere

1 Anomalous Dielectric Constant ofArtificia1 Ionosphere (1937) 2 On the Tbtal Reflection of Electromagnetic Waves in the Ionosphere (1938)

Various theories of the propagation of radio waves through the ionosphere were developed during 1930-1938 based on either the ray treatment or the wave treatment. The refractive index comes out in general t o be a complex quantity which is a function of the electron number density and collision frequency which are functions of height. Consequently, the solution of Maxwell's equations becomes highly complex. Appleton deduced certain conditions of reflection by assuming that the refractive index must vanish.

However, since the refractive index is complex, other criteria were formulated by various workers (D R Hartree, Proc Camb Phil Soc 25,47,1929: ibid. 27 143,1931; Proc RoySocA131,428,1931; Fosterling and Lassen,Annd Physik 18,26,1933; H G Booker, Proc Roy Soc A 165,235,1936; M N Saha and R N Rai, Proc Nut Inst Sci Ind 3,359, 1937; M N Saha, R N Rai and K B Mathur, Proc Nut Inst Sci Ind 4,53,1938). R N Rai (Proc Nut Inst Sci Ind 3, 307, 1937) suggested that a better criterion of

reflection would be that the group velocity of the wave vanishes. This gave him a new condition in addition t o those of Appleton, a condition that was experimentally confirmed by Pant and Bajpai (Science and Culture 2, 409, 1937), L Harang (Zkrr Mag 40, 29,1937) and R Jonaust, Abadie and Joigny ( L'onde electrique 16, 185, 1937).

Experiments also indicated that apart from totally reflecting electromagnetic waves the ionosphere also partially reflected and partially transmitted these waves. More-


over, for very long waves, the complex refractive index can change within one wavelength and reflection can occur even though the refractive index is not zero. These features could not be explained by any of the theories.

Both Appleton's and Rai's conditions of reflection were obtained by neglecting the effect of damping, a n essential feature of the physical process. The questions that arose were: (1) do both these conditions follow as consequences of the Maxwell-Lorentz theory and (2) what are the conditions of reflection when collisional damping cannot be neglected? These are the problems that Bose set about to tackle in the 1938 paper.

Instead of Maxwell's equations with a complex refractive index, Bose used the microscopic equations of the Maxwell-Lorentz theory and the method of characteristics used by Hadamard, Debye and others. The method is general and can also be used for the propagation of light in a material medium. However, Bose's results were expressed in symbols unfamiliar to workers in ionospheric physics. I t was M N Saha and K B Mathur ( Ind Jour Phys 13,251,1939) who gave a critical assessment of Bose's results in a form easily comprehensible to them. They showed that Bose's treatment gives the same results a s those of previous workers for the case of vertical propagatipn. However, in the presence of collisions the nature of the complex refractive index and polarization became complicated, and Bose's conditions do not give accurate results.

In 1960 H K Sen and AA Wyller ( J Geophys Rev 65,3931,1960) used the Chapman- Enskog method and the Baltzmann equation to obtain the effect of the velocity dependence of the collision frequency in a closed analytic form.

Earlier in 1937 Bose and S R Khastgir had investigated the conditions under which the value of the dielectric constant of an ionized medium exceeds unity (anomalous behaviour) and carried out simple experiments with ionized air in a discharge tube (artificial ionosphere) to show that the dispersion formula alone cannot explain the observed anomalies.

EDITORS

Mathematical Physics

1 Studies in Lorentz Group.(1939) 2 The Complete Solution ofthe Equation:

3 On an Integral Equation Associated with the Equation for Hydrogen Atom (1945)

The three papers are on various problems in mathematical physics: the first deals with the properties of the matrix group 0 (4,C) in its defining representation, with consequences for the physical Lorentz group SO' (3,l); the second develops a particular technique, involving integration in the complex plane ii la Sommerfeld, for solving the inhomogeneous Klein- Gordan equation; and the third is concerned with the Schrijdinger equation for the Coulomb problem, analyzed as an integral equation in momentum space. These papers were written in 1939, 1941 and 1945 respectively A few introductory comments regarding each are presented.

It is most easily seen from the infinitesimal approach, provided that one works with complex linear combinations ofthe basis elements, that the ~ i e algebra of SO(3,l) formally splits into two commuting SU(2) or angular momentum like algebras, This helps in the construction of all the finite dimensional irreducible matrix representations of SO(3,l), and the connection of SL(2,C) also becomes clear. One sees on the way the possibility of formally expressing a finite element as a product of two commuting factors, in the complex sense. To realize all this without using inhitesimal techniques, however, requires considerable ingenuitx This is what is attempted in the 1939 paper of Bose. He in fact examines the

2 2 S N Bose : The Man and His Work

much larger group of all complex orthogonal transformations in four dimensions, namely the group O(4,C). The key idea is to deal with symmetric elements of 0 (4,C) and find ways of characterizing and factoring them; and for antisymmetric matrices, exploiting the properties of self-dual and antiself-dual sets of matrices. It is the latter that actually lead to the commuting structure referred to above. From the discussion of O(4,C), Bose descends to the case of SO(3,l) by imposing the requisite reality conditions, and thus connects up the spinor approach based on SL(2,C), as typified in the familiar s tatement (112,112) = (1/2,0) x (0,112) : the four-vector representation of the Lorentz group is the product of the two basic spinor representations.

Turning to the second paper, it is basically an exercise using relativistically invariant solutions of the Klein-Gordan equation to solve a problem for which H J Bhabha in an earlier paper had presented a solution in a particular form. At the time the paper was written, namely 1941, the technology associated with the wave equation using Stiickelberg- Feynman functions, invariant Green's hnctions, retarded and advanced ones, etc., were presumably not yet widely known and used. The interest in Bose's paper lies in its use of complex contour integration methods, a favourite of Sommerfeld, to get solutions of the inhomogeneous Klein-Gordan equation in Kirchhoff-like form.

The third (1945) paper is devoted to a discussion of the quantum mechanical Coulomb problem in momentum space. It has been known for a very long time, through the work of V Fock and V Bargmann, that if the Schrodinger equation for energy eigenfunctions is expressed in momentum space, it has the form of an integral equation, and moreover i t makes the higher symmetry in the problem manifest. Thus the SO(4) symmetry for bound states, and S0(3,1) for scattering states, can be explicitly exhibited by suitable energy- dependent choices of variables. (It is curious, however, that Bose does not refer to Fock's and Bargmann's papers a t all). Bose however proceeds somewhat differently realizing that the energy-dependent choice ofvariables would discriminate between bound and scattering states. He exploits methods of Hobson involving solid harmonics and operator calculus to explicitly solve the integral equation, after reducing it essentially to a radial problem (in momentum space). An interesting feature of his expressions is that the quantization of the bound state energies, and also the fact that the angular momentum quantum number is bounded above essentially by the principal quantum number, both arise from the require- ment that the momentum space eigenfunction be single-valued, rather than that i t be normalizable. I t is the latter condition that is normally used in a configuration space solution to the problem. His method of course leads to both bound and scattering state wave functions. At the end of the paper, a completeness statement involving the former alone is developed. One presumes therefore that this is distinct from the usual physical notion of completeness of eigenfunctions of the Hamiltonian operator, since that would have brought in the scattering states also.

All three papers show a taste and knack for clever algebraic and analytic manipulations, even though the problems addressed are circumscribed in scope.

N MUKUNDA Indian Institute of Science, Bangalore.

Unified Field Theory

Les identitds de divergence duns la nouvelle thdorie unitaire (1953) [30.3.531 Une thdorie du champ unitaire auec T,, # 0 (1953) [18.7.531 Certaines consequences l'existence du tenseur g duns le champ affine relativiste (1953) [18.7.531 The affine connection in Einstein's new unitary field theory (1954) [ 29.9.52 I Solution d%ne dquation tensorielle intervenant duns la thdorie du champ unitaire (1955) [Receipt date not mentioned]

The papers are listed in chronological order. The journal receipt date is given within brackets.

The general theory of relativity established itself as a satisfactory description of gravitation. The only other interaction that was then known was the electromagnetic, and a natural sentiment was to bring the electromagnetic and gravitational interactions under one umbrella -to be more precise, to geometrize the electromagnetic field as well.

Einstein, however, demanded much more from a unified field theory. He was committed to the field idea and was never reconciled to quantum mechanics with its basic indeterminacy. He expected that matter itself would ultimately resolve into a field and, as singularities mean a breakdown of the field concept, there should be no singularities in a truly unified field theory. Again, when Kaluza's five-dimensional geometry appeared, the idea was that our observations were limited to four dimensions


and thus were essentially of an incomplete nature. One might wonder whether this incompleteness could account for quantum indeterminacy.

What, by the way, is a unified theory ? We may quote Einstein's definition : Neither the field equation nor the Hamiltonian can be expressed as a sum of several invariant parts but are formally unified entities.

There were a host of unified field theories - one may compare the ever-changing theories with the changing models of cars. But they all had one similarity. The 4-dimensional Riemannian geometry has ten independent variables (the metric tensor components gih ), just sufficient for the gravitational field whose source, the energy-. momentum-stress tensor, has also ten components. So, to incorporate the electromagnetic field, one must have some additional geometric variables. These additional variables were introduced in a number of ways, e.g.

(a) In the Weyl theory, i t was postulated that the norm of a vector changes in a parallel displacement according t o the law.

and the vector Qi was identified with the electromagnetic potential vector. (b) In Kaluza's five-dimensional formalism the metric tensor has 15 components. (c) The fundamental tensor gik was assumed to be complex, so that there were 20

variables. (dl The fundamental tensor was assumed to be non-symmetric and the affinities

T which define parallel displacement were also taken to be non-symmetric. There were thus altogether 80 variables.

Over a number of years different forms of the non-symmetric theory were developed by Schriidinger and Einstein. One particular form gained wide publicity as i t was announced by Einstein first in the popular press in December 1949 and then presented in the 1950 edition of his book Meaning ofRelativity as an Appendix. In his papers Bose used the adjective 'new'before unified field theory without indicating specifically which particular theory he had in mind. It seems clear that he referred to this theory.

We give a table to pinpoint the salient points of this theory as compared to the general theory of relativity :

G T R U F T 1. Symmetric metric tensor g,,, and symmetric Non-symmetric g,,, and T; ,. [ What is the

affinities r ,. metric tensor ? No specific statement is made in the book but it became common to identify the symmetric part of g,,, with the metric tensor and the antisymmetric part was related to the electromagnetic field, as we shall presently see. ]

Unified Field Theory

(i.e. gpv,a-gpv Ga-gpp I P v = 00). [ The.solution of these 64 equations, i.e. expressing r ' s in terms of g,,, and their derivatives, was a n extremely difficult problem and was given independently by Bose, Einstein and Kaufman, Hlavaty and Tonnelat. Later there were also other workers in the field including R S Mishra.]

3. Field equations obtained from the variational Field equations obtained from the variational principle principle

where R,, , (the Ricci teneor) is f l 's & r 'S varied independently. The tensor

R ~ ~ = ~ ~ ~ - ~ ~ & + ~ ~ , - ~ ~ , ~ = R ~ . uikis 1 The variation can be performed in two ways :

uik = R~~ - [ri, - rkSi) + ri r,] (1) g% are to be varied and the relation

between Ps and g,,,'s assumed. Ti = I'b and

(2) p P s are independently varied-the heik = r if r :k + + r is, i] - r ; r - r $, - - relations between P s and g,,,'s appear as one set of field equations.

where underlining indicates the symmetric part and the hook the antisymmetric part. The tensor uik is selected by some ad hoc conditions introduced by Einstein which may be summarize$ in the form Uki = Uik (r) uik = Rik ( A ) where

A$ = 0. The conditions were not justified by

any theoretical consideration. Bose therefore sought to work with a different variational principle. However, this led to a much more complicated set of equations which involved two arbitrary constants.


Field equations R w = 0 .

Field equations R i k = 0, (a)

The equations satisfy four differential g i k , 1 = 0 , t

(b) identities. ri = 0 . (c)

The equation Ti = 0 was introduced ad-hoc on the ground that this does not cause an over- determination. The equations (b) and (c) together give ( G g v ) , k = 0. This led

Einstein to identify g@ with the dual of the electromagnetic field tensor-there is no free magnetic pole.

If fields other than gravitational were present, The electromagnetic field is already taken care additional terms were introduced in the of. No mention is made of other fields. Lagrangian and the field equations then Unification is thus not complete. became

Physical conclusions drawn all agree with No physical conclusions could be drawn. observations. Indeed, Pauli remarked (1958), Whether the

field equation. . . can actually be connected with physics a t all is rather doubtful.'

Before going over to the subject matter of these papers, i t is somewhat interesting to note that a t least four of these papers were produced within the span of less than a year. That was something unusual in Bose's career. Never before had he written so many papers in a year and never before had he contributed as many as five papers in a single field. One may wonder whether these works kindled Bose's enthusiasm to a degree he had never experienced in his whole life. And mark, five papers in four different journals -was it all fun with Bose !

The first paper concerned the divergence identities in the unified field theory we have just outlined. Long before, Hilbert (1924) had proved that the set of equations obtained from a variation principle would not be independent but there would be four divergence identities satisfied by them. For non-symmetric theories Einstein obtained the identities by a method which was rather complicated. Bose showed that they could be obtained in a somewhat more general manner by following Hilbert's procedure.

In the second paper Bose used a much more complicated Lagrangian. Einstein's Lagrangian was obtained on the basis of somewhat arbitrary arguments and then the equation Ti = 0 appended, justifying this by the rather curious reason that i t would not make the equation set overdetermined. Bose's equations were naturally more complicated and were not supplemented by the ad hoc condition Ti = 0. However, the equation system involved two arbitrary constants and hence they were not quite appealing. Neither Bose nor any of his students investigated these equations.

Unified Fisld Theory

THE INSTLTUTE FOR ADVANCED S T U D Y PRIXCCTON, NEW J C R S I T

Profes9or S.N.Eose Univsrsi ty P&lless g f Science 92, Upper Circular Road Calcutta,9, India.

Dear Professor Bose:

Thank you f o r your l e t t e r ol S ptember 20th. I am led to sea t h a t you a r e interested i n t h i s tKeorI and t h a t you have dovoted so nluch.work and penetration to the s o l u t i o n o f the equation

p + c =r t -

I bel ieve , to be sure, t h a t the ~ o l u t l p n o f those equations i s not o f g r e a t help toward the answer o f the question: Uo the s l n p l e r j ty - f ree so lu t lons o f the equat ion aystem have physical meaning? Are them a t a l l s ingu ln r l ty - f ree so lu t ions whi dh correspond to the a,torr.ls t i c chnt*ric t o r o f ' matter end radiation? I t snems to w t h a t the rnathematicnl methods ava i l ab le a t present a r e not powerful enouch to an?lwer t h i s question.

!rowever, I am now flrrnly convinced t h n t the equ? tion-sys tern

j t q ; 4 = 4

fl* =o represen t s the only r e l a t i v i s t i c law o f

r=o- &A,< v + &,, + Rt:,a = O

formally & t u r a l ~ e n s r a l i z a t i o n o f the e r a v l t a t i o n .

With k i n ? regards,

s incere17 yours,

4 ~~~~. Alber t E ins te in .

Einstein's letter of 4 October 1952 to Bose


Unified FieM Theory

* %&r- $4 .,r-$24 3~ G r L L ; - " L yYW A.2.

Einstein's k t e r to Bone in Paris from Princeton, dated 22 October 1953, commenting On Base's paper 'A U ~ t a r y Field Theory. . ' for Le Journal de Physique; by cour+esy of The Jewish National and University Library, Jerusalem.


The remaining three papers are all on the equation

In the third paper Bose considered the above equation as first order partial differential equations for g,. Then the condition of integrability is g,,, ,p = g,,, s,. This condition reduced to equations involving r's and g,, but free of the derivatives of g,,. They were explicitly worked out by Bose. This was a more or less routine investigation and did not lead to any very impressive result.

In the last two papers in the list Bose was interested in solving the above equations, regarding them as coupled algebraic equations for T's, taking the g,,'s as known. The problem is to find the solution not for particular coordinates but in a form which will hold in all coordinate systems. This was then a challenging problem especially in view of a remark by Schrodinger (1947) :

In the general case i t is next to impossible to give a tensorial solution of these equations.

The adjective 'tensorial' in Schrodinger's remark is not quite correct for T's are themselves not tensors ; what he meant was that the solution must hold in all coordinate systems. However, Schrodinger was soon proved to be unduly pessimistic. Mme Tonnelat, Hlavaty in 1953, Bose, Einstein and Kaufman in 1954 and later on many workers including Mishra gave solutions. Although Bose's paper was published somewhat later than the first papers of Tonnelat and Hlavaty, Bose's first paper was communicated in September 1952, earlier than the publications of Tonnelat and Hlavaty. Indeed, all three were ignorant of the works of others and their methods were also different. Again, while Einstein had received a pre-publication manuscript from Bose and was aware of the work of Hlavaty, he and Kaufman proceeded without splitting g,, and Pkl into their symmetric and anti-symmetric parts, unlike the procedure adopted by Bose and others. Strangely, some later workers like Mavrides, Kichens- samy and even Mishra (1976) comment on the works of Tonnelat and Hlavaty, but they seem to ignore the works of Bose and also of Einstein and Kaufman.

IMPORTANCE OR MOTIVATION OF THESE WORKS

One may wonder what one can gain by these solutions. The first answer is that a pure mathematician's motivation arises simply from the mental desire to solve a problem just as the mountaineer climbs the Everest simply because 'it is there'.

Perhaps some of the eminent mathematicians had the idea that once the T's are found in terms of g,,, these values of T would be plugged into the expression for R,, and one could attempt a general solution of the equation R,, = 0. However, even in the much simpler case of general relativity, such a general solution of R,, = 0 cannot be obtained and in reality has not even been attempted. What, therefore, one has to do is either to assume some symmetry and in that case the calculation of T's becomes not that difficult (and has been done by Bonnor, Papapetrou and others), or to deduce

Unified Field Theory 3 1

some general theorems. Thus to Einstein the crucial problem was: 'Do the singularity- free solutions of the equation system have physical meaning ? Are there a t all singularity-free solutions which correspond to the atomistic character of matter and radiation ? From this viewpoint the solution of those equations is not of great help. [ Here those equations refer to the equations whose solution Bose presented in a letter to Einstein and we have just now considered. I

But, if the solution of the T's was not important from the point of view of the theory, one may ask: why did then Einstein and Kaufman give a solution ? The fact is, i t came as a by-product of their investigation of the following problem. The universe we observe has three space-like and one time-like dimension. So, in order to be physically accept- able, the signature must be 2 or, in other words, the metric tensor determinant must be negative. Einstein and Kaufman considered provi~ional lyg~ to be the metric tensor and then investigated if this has the correct signature - do the field equations ensure that this signature will be maintained everywhere ? Their answer was in the affirm- ative, and in course of the proof they found the solution of the equations gik 1 = 0.

As we come to the close of our discussion of Bose's works in-unified field theories, some disturbing questions come to mind. Did Bose share Einstein's attitude towards quantum mechanics ? Did Bose believe that a viable unified field theory based on geometry can really come about while most physicists thought these attempts were futile ? Bose has left nothing in writing which can throw any light on these matters. In fact, his papers were purely mathematical exercises without any reference whatso- ever to physics.

A K RAYCHAUDHURI Former Professor,

Presidency College, Calcutta

Thermoluminescence

A Report on the Study of Thermoluminescence (1955)

Thermoluminescence means the phenomenon of emission of light (infrared or ultraviolet radiation included) other than pure thermal radiation, by a system under thermal stimulation. I t is evidently related to energy states of the physical system, the relative transition probabilities and related electronic processes. When a system is excited by any method and a part of the excitation energy is stored in it, the system-is termed thermoluminescent if it releases, on heating, a part or whole of the stored energy in the radiant form. The phenomenon of thermoluminescence exhibits a very large variety of behaviours, corresponding to different substances with variations in impurity contents or other imperfections. Thus, it is not surprising that thermoluminescence in all i ts details is a comparatively poorly understood phenomenon even in the case of simple materials.

Thermoluminescence appears to have been first observed by Boyle about three hundred years ago. Since the publication of Randall and Wilkins' work in 1945 there has been a steady accumulation of data in this field a s well as attempts a t theoretical interpretation of the experimental results in the light of the modern theory of solids, which already met with striking success in allied fields. The applicational p~ssibilities of thermoluminescence in geological and archaeological work, dosimetry, or as a research tool in science and industry, have been already demonstrated by the tremendous growth in the number and variety of work in this field in the last fifty years.

Thermoluminescence 3 3

Although extensive work has been carried out with thermoluminescence as a research tool, most of such work consists of the determination of trap depths and their changes under varying experimental conditions, but the results are far from satisfactory for a clear understanding of the electronic processes occurring inside the solid. The identification of the trapping and emitting centres, responsible for the different glow peaks, remains as yet a challenging problem in most of the phosphors. It was felt by Bose ( and other pioneers in the field) that the physical interpretation of the results of investigations on thermoluminescence would be facilitated if the spectral compositions of the thermoluminescence and afterglow emission could be compared with the luminescence of the phosphor under study The research workers in this field had been trying t o find out the spectral compositions of thermoluminescence and afterglow emission with the help of filters; naturally, the results were indicative of the fact that the spectral composition of the emission undergoes changes during the process of therrnoluminescence or afterglow decay in many cases but the data were not of convincing precision.

The duration of thermoluminescence is generally of the order of a minute and afterglow emission, comparatively poor in intensity, may also change in spectral composition as the intensity changes with time. The difficulties of interpretative work were thus essentially due t o the lack of a suitable measuring device and experimental techniques. Bose saw the crux of the problem, and designed a rapid scanning spectrophotometer of comparatively high sensitivity to meet the requirements of the experimental workers in th i s field. The design and fabrication of the scanning spectrophotometer was reported by Bose a t the International Conference on Crystal- lography held in Paris in 1954.

The spectrophotometer is capable of scanning the entire spectral range of visible and near-ultraviolet radiation in 0.9 seconds, and the spectral distribution of emission either in afterglow decay or therrnoluminescence can be displayed on the screen of a cathoderay oscilloscope. It was thus possible t o observe the changes in the spectral composition of emission during any time-varying process like thermoluminescence. The instrument performs the dual role of a spectrograph and a microphotometer, and can very conveniently be adapted t o carry on studies of (i) fluorescence and phosphorescence, (ii) thermoluminescence spectra, (iii) the temperature dependence of luminescence (both fluorescence and phosphorescence), (iv) the correlation of colour centres and other known trapping centres with luminescence, (v) the development of emission centres in the phosphor during X-ray and cathoderay irradiation, and (iv) near-infrared absorption and emission spectra etc. The possibilities of the scanning spectrophotometer have however not been fully exploited till date.

The special features of the work initiated by Bose a t the Khaira Laboratory in Physics in Calcutta were ihe luminescent and related properties of solids and frozen liquids under excitation by low energy electron beams as well as X-rays. The excitation is limited to a thin surface layer in the case of irradiation by low energy electrons, while for X-rays or other high energy excitations the entire volume of the crystals is affected. The two methods of excitation should thus bring out the distinctive differences in the


effects of impurities , self-absorption and energy transfer processes, on the luminescent and allied properties of solids. Some work was also attempted on the luminescence of solids under soft X-ray excitation; and attempts were made to correlate the data on valence band spectra with those on colour centres and luminescence of simple solids.

Adapted from H N Bose, On the Pioneering Works of Professor S N Bose in the Field of Thermoluminescence, in the Proceedings of the Seminar on The Scientific Contributions of Professor S N Bose, Cal Math Soc 1943, pp. 111-121.

Bose Statistics : a historical perspective

a. Planck's derivation of his law of black-body radiation In order to appreciate the novelty and importance of Bose's work in its historical perspective, it is necessary first to give a brief account of Planck's original derivation [Planck, 14 December 19001 of the law of black-body radiation,

which he had discovered empirically a few weeks earlier (7 October 1900). I t consisted of three steps.

First, he established the relation

between the energy density pv of incident radiation a t temperature T, whose frequency lies between v and v +dv, and the average energy Uv of a resonator of the same frequency v a t temperature T, on the basis of classical electromagnetic theory Com- paring (1) and (2), he found Uv :


In the second step he determined the entropy of these oscillators by integrating TdS=dU where T is taken from (3) as a function of U (for fixed v). He obtained

The third step was the revolutionary one. He introduced two ideas a t this stage that he himself considered a s 'acts of desperation'. He assumed that the total energy UN = N U, of N oscillators (resonators) was made up of finite energy elements E such that UN = PE with P a large number. Then he searched in Boltzmann's work for a permutation measure WN (the total number of complexions or distributions)'of P discrete equal energy values E among N oscillators that would correspond to the right hand side of (4) when used in SN = k In WN. He found

Using P / N = U, /E , SN = N S and applying Stirling's formula, he obtained

Since S is a function of (U, /v) only, i t follows from (4) and (6) that

This is how quantum theory was born. Planck had no more justification for using the formula (5) than that it gave him the result that he was after : '. . . a theoretical interpretation had to be found a t any price, however high i t might be,' he wrote to Robert William Wood in October 1931. [Mehra and Rechenberg, 1982,1,1,501. Actually, Planck's combinatorial approach differed from Boltzmann's probabilistic method in tha t Planck associated WN with SN, the equilibrium entropy, without maximizing it. For Boltzmann WN was the number of possible complexions corresponding to the macrostate which can be realized by the largest number of complexions. As pointed out by Ehrenfest, a strict adherence to the accepted principles of statistical mechanics would have led Planck to the classical law of Rayleigh and Jeans! [Ehrenfest, 19051

b. Einstein's light-quantum hypothesis In 1905 Einstein argued on the basis of his analysis of energy fluctuations of radiation obeying Wien's law that such radiation had statistical properties similar to material particles and so must consist of discrete light-quanta of magnitude ( R / N , ) pv = hv. [Einstein, 19051 This light-quantum hypothesis was immediately applied by him to give a reasonable explanation of several radiation phenomena, specially the photo-electric effect. Since these conclusions contradicted the classical electromagnetic theory on which Planck attempted to base his theory of heat radiation, Einstein considered

Bose Statistics : a historical perspective 3 7

Planck's theory 'in some way a counterpart'to his own theory. He subjected Planck's derivation to a critical analysis and came to the following conclusion :

We must therefore regard the following law as the basis of Planck's quantum theory of radiation : the energy of an elementary resonator can only assume values which are integral multiples of (R /No) pv ; the energy of a resonator changes in jumps by absorption or emission in integral multiples of (R /No) pv. . . . If the energy of a resonator can alter only in jumps, then for the evaluation of the average energy of a resonator in a radiation cavity, the usual [electromagnetic] theory cannot be used, for the latter does not admit any distinctive energy values for a resonator. [Einstein, 19061

Two years later Lorentz also came t o the same conclusion [Lorentz to Wien, 6 June 1908; Mehra and Rechenberg, 1982, 1, 1, 981. Planck attempted to modify his theory to take into account the quantum of action without violating any aspect of classical electrodynamics, but i t gradually became clear from his studies and those of others that it was impossible to establish the theory of black-body radiation entirely on the foundation of Maxwell's classical electrodynamics and the statistical mechanics of Maxwell and Boltzmann.

c. Debye's derivation of Planck's law In 1910 Debye gave a new derivation of Planck's radiation law in which he sought to avoid the inconsistencies (as viewed from Maxwell's theory) of Planck's earlier derivations as pointed out by Einstein [Debye, 19101. Instead of using the relation (2) between the radiation density p, and the average energy of the oscillator Uv, Debye calculated 'the probability for a given state of radiation and therefore as is well known, the entropy, using the properties of the state alone without employing resonators.' In agreement with Rayleigh and Jeans, he calculated the number N, dv of elementary states or vibrational modes (Hohlraum oscillators) contained in a volume V and obtained

Assuming that an amount of energy hv gets distributed over each of these vibrations according to an arbitrary distribution function fv , he obtained

Now, 'analogously t o Planck', Debye assumed that the probability of distributing N, f, dv quanta of energy hv among Nv dv vibrations was given by

(N, dv +Nv f, dv) ! O v =

(N, dv) ! (N, fv dv) !

He then calculated the distribution function f, in the following way. He calculated the


maximum of w, with the constraint that the energy remained constant. From this he calculated the equilibrium entropy S , to be k In (o 7" ). Finally, using the definition

of entropy to be (dSv/dpv) = T ', he obtained

Together with (9), this yielded Planck's radiation formula (1). 'hvo important features of Debye's derivation need to be emphasized as bwkground

to Bose's derivation that came fourteen years later. First, i t became clear from his derivation that Planck's law follows simply from the assumption that the energy transfer from matter to the heat radiation and vice versa is quantized in units of hv and that no knowledge of theproperties of material resonators is needed. In other words, the quantum hypothesis only determined the transfer of energy from one mode of the radiation to another. This was in keeping with Planck's ideas but not Einstein's. Second, he used Planck's definition of the probabilities a,, without analyzing its significance.

d. Indistinguishability of Planck's quanta In 1911 Ladislas Natanson [I9111 subjected the Planck-Debye combinatorial procedure to a critical analysis and showed that it was equivalent to distributing P indistinguishable energy elements E among N 'receptacles of energy' distinguished only by the numbers j of quanta in them so that Nj receptacles contain j quanta each, subject to. the constraints Z Nj = N and Z jNj = P. He found

j j

(Nv dv) ! W v =

P

Contrary to popular belief, therefore, the indistinguishability of the quanta had already been tacitly assumed by Planck and Debye and this was first noticed by Natanson already in 1911.

Ehrenfest and Kamerlingh Onnes [I9141 also came to a similar conclusion in 1914. They however claimed that Planck's procedure was equivalent to distributing P indistinguishable energy elements E among N distinguishable resonators and arrived a t the expression

(N- l + P ) ! W v =

P ! ( N - I ) !

which is equivalent to the expressions (10) and (12) when N>>l. They then proceeded to clarify the distinction between Planck's energy quanta which were statistically not independent (they called them 'non-disjointed quanta') and Einstein's hypothetical light-quanta which were believed to be statistically independent of one another ('dis-

Bose Statistics : a historical perspectiue 39

jointed quanta'). The concluded : 'Planck3 formal device (distribution of P energy-elements E among N resonators) cannot be interpreted in the sense of Einstein's light- quanta. '

e. The introduction of phase space cells by Planck Planck himself had come to the view on his own that classical statistical mechanics had to be modified in order to yield his radiation formula (1) rather than that of Rayleigh and Jeans. He reported his important conclusion a t the Solvay Congress held in Brussels in 1911. According to Gibbs, the probability of finding a single particle in the element dS p d3 q of the six-dimensional phase space is given by

where E= E(p,q) is the energy of the system. For a classical one-dimensional oscillator, E = p2/2m + P q 2 / 2 and so the average energy of such an oscillator is kT. However, if E can take only the discrete values En = n E = n h v with n=0,1,2 . . ., then

in agreement with (3). I t was this straightforward derivation that led Planck for the first time to make the explicit statement that the energy of an oscillator (and not only its average energy) was an integral multiple of hv. It also led him t o interpret the constant h as a finite extension of the elementary area in phase space. The motions of one-dimensional classical oscillators can be described by a family of concentric ellipses

'h '/z of semi-axes (2E43) and ( 2 m E ) . Since these oscillators can have any energy E, the ellipses form a continuum. However, if it is assumed that these ellipses are separated from each other so that the nth ellipse encloses an area

then the energies of the oscillators must be integral multiples of hv. Planck therefore drew the revolutionary conclusion that energy quanta were a consequence of the fundamental condition

This implied a revision of the basic tenets of classical mechanics. He declared : 'The


framework of classical mechanics, even if combined with the Lorentz-Einstein principle of relatjvity, is obviously too narrow to account for all those physical phenomena which are not directly accessible to our coarse senses . . . One should therefore confine oneself to the principle that the elementary region of probability h has an ascertainable finite value and avoid all further speculation about the physical significance of this remarkable constant.' In other words, all attempts a t finding a classical mechanical explanation of h must be abandoned. As we shall see later, to Bose these considerations (with which he was familiar) brought home extremely important lessons that would influence his own seminal contribution to the subject. First, he took serious note of the conclusion that classical eledjrodynamics was essentially incompatible with Planck's radiation formula. Second, since the quantum states of a particle were not continuously distributed throughout phase space (as in classical dynamics), he concluded that their number could be counted by simply dividing the total volume of phase space by h3, the volume of an elementary cell. Finally, he took seriously Planck's claim that classical statistical mechanics had ta be modified in a special way in order to conform to the requirements of quantum theory.

f. Spontaneous and induced transitions The next important step was taken by Einstein in 1916. In 1913 Niels Bohr had proposed his model of the atom with stationary states with discrete energy values [Bohr,1913, b,cl. The transitions between any pair of these states were assumed to be caused by the exchange of energy quanta hv between the atom and the radiation field described by classical electrodynamics. Einstein gave an elegant derivation of Planck's radiation formula by considering Bohr atoms in thermal equilibrium with radiation in a cavity but without making any use of classical electrodynamics [Einstein, 1916al. He used Boltzmann's principle to write the probability Wn for an atom to be in a stationary stat9 with quantum-number n in the form

gn being the statistical weight of the state. He then assumed that a stationary state n may pass to a stationary state m of higher energy (E, > E, ) by absorbing a light- quantum of frequency v, , , the rate of transition being B r ATn p, (Nn being the number of atoms in the stationary state n and Br a proportionality constant characterizing the absorption of radiation of frequency vn ,in the presence of external radiation of density p,). Transitions from a stationary state m of higher energy to a stationary state n of lower energy by the emission of a light-quantum of frequency v,, = v,, could, however, take place in two ways. Atoms can emit this radiation independent of the external field, their number being A: N, (A: being the constant of proportionality). About this Einstein said : 'One can hardly think of i t in any other way except as a radioactive reaction.' I t came to be known later as 'spontaneous emission.' Atoms can also be induced to emit radiation by the external field, the rate being B: N,,, p,. Einstein


called th i s 'negative radiation'. In thermal equilibrium one mus t have (A: Nm + Bk N, pv) = B: Nn pv . Using (18) to calculate N,,lN,,,, one gets

At this stage Einstein had to take his guidance from classical theory by essentially applying the 'correspondence principle' according to which classical theory should be a limiting case of quantum theory. In the classical limit p, goes to infinity as the temperature T goes to infinity for fixed v (RayleighJeans law, p, aT). This gave BL g, = BF gn. With this condition ( 19) could be written as.

A; / Bk Pv = - exp [ ( e m - E , ) / k T ] - 1 :

Then Einstein used Wien's displacement law to infer that Ak/Bk must be proportional to 3, and the Rayleigh-Jeans law in the limit of low frequencies to determine the constant of proportionality t o be 8 x h/c3 . Thus he obtained the relation

and

which is Bohr's frequency condition. These two relations when substituted into (20) gave Planck's law. This is a remarkable derivation that uses an admixture of the light-quantum hypothesis and Bohr's atomic theory which Bohr regarded as being imcompatible, Wien's radiation law and therefore implicitly the Maxwell-Boltzman distribution for the light-quanta, and the correspondence principle which was a heuristic principle without any fundamental basis. Einstein concluded by saying : 'I admit freely, of course, that the three hypotheses concerning outgoing and incoming radiation, do not a t all become substantiated results by the mere fact that they lead to Planck's radiation formula. But the simplicitly of the hypotheses, the generality with which the consideration can be carried through easily, as well as the natural connection of the consideration employed with the limiting case of Planck's linear oscillator (in the sense of classical electrodynamics and mechanics), persuaded rrie to regard i t as very probable that all this constitutes the fundamental outline of the future theoretical derivation.' Although the hypotheses concerning the outgoing and incoming radiation turned out to be correct to a large extent (though not entirely so, as we shall see later), the future derivation (given by Bose) did not inake use of them a t all. In fact, they became the


major bone of contention between him and Bose. By 1916 the light-quantum hypothesis had received a measure of empirical support

from Millikan's verification of Einstein's photo-electric equation. Millikan's paper [1916b] appeared in the March 1916 issue of the Physical Review. Einstein quickly realized from his new derivation of Planck's law that atoms seemed to interact with radiation as if single atoms collided with light-quanta of energy hv and directed momentum hv lc . He wrote to his friend, Michael Besso :

The fundamental thing is that the statistical consideration, which leads to Planck's formula, has become unified and thereby as general as one can imagine, since one has not assumed anything about the nature of the mediating molecules other than the most general quantum idea. From it follows the result (which was not contained in the paper I sent you) that in each energy transfer from radiation to matter the momentum hvlc is also transferred to the molecule. Hence we conclude that every such elementary process is a completely directed event. With that the light-quanta must be considered as being substantiated. [Einstein to Besso, 6 September 1916; Mehra and Rechenberg, 1982,1,2,515]

However, Einstein was not really satisfied. In a subsequent paper he remarked : The weakness of the theory lies, on one hand, in the fact that it does not bring us closer to a connection with the undulatory theory, and, on the other hand, in the fact that i t leaves the instant and direction of the elementary processes to chance. [Einstein, 1916b, 19 171

g. Reactions to the light-quantum hypothesis: discovery of the Compton effect

The majority of physicists, in fact, did not take the light-quantum hypothesis seriously, even after Millikan's work. Millikan himself remarked in his 1916 paper on the photo-electric effect : 'Yet the semi-corpuscular theory by which Einstein arrived a t this equation seems a t present to be wholly untenable.' The main difficulty lay, as Einstein himself realized, in accounting reasonably for the interference and diffraction phenomena observed with all kinds of radiation in terms of light-quanta. Evidence in favour of the wave theory was overwhelming and mounting. In 1912 Max von Laue, Walter Friedrich and Paul Knipping had settled the issue in favour of X-rays being electromagnetic waves of extremely short wavelengths through the phenomenon of interference of X-rays in crystals [von Laue, Friedrich and Knipping, 19121. In 1913 Planck, Nernst, Rubens and Warburg who wished t o bring Einstein t o Berlin, wrote to the Prussian Ministry of Education:

That he may sometimes have missed the target in his speculations, as, for example, in his theory of light-quanta, cannot really be held against him. For in the most exact of natural sciences every innovation entails risk. [Clark, 1972, 2151

Bohr had his own reasons for not favouring the light-quantum hypothesis though he realized from the beginning that the laws of electrodynamics had to be abandoned

Bose Statistics : a historical perspective 43

for a proper description of quantum phenomena. The hypothesis ran counter to his plans to build atomic theory on the basis of the correspondence principle,

The general acceptance of the light-quantum hypothesis eventually came in 1923 through the work of Debye [I9231 and Compton f19233. Compton realized that the shift in the wavelength of X-rays scattered by atoms that he observed could not be understood in terms of the classical wave theory of scattering but only in terms of elementary processes of scattering of light-quanta by the electrons in atoms, in which both energy and momentum were conserved. Debye also arrived a t the same conclusion independently. .Arnold Sommerfeld who was then visiting the University of Wisconsin as Carl Schurz Professor of Physics came t o know about Compton's results. On 21 January 1923 he wrote to Bohr : 'The most interesting scientific news I have learned in America is a work of Arthur Compton in St. Louis. According t o i t the wave theory of X-rays would finally have.to be given up.' [Mehra and Rechenberg, 1, 2, 5291 Wherever he went, Sommerfeld referred t o Compton's work and its importance for the quantum theory. On his return t o Germany he continued to advocate Compton's work. Several months later he wrote to Compton : 'Your discovery of the change in wavelength of X-rays also occupies the scientific community in Germany most deeply. I met Einstein and Kossel in August, and we mainly discussed your effect;' [Sommerfeld t o Compton, October 1923; Mehra and Rechenberg, 1982,1,2,5301

h. Pauli's derivation of Planck's law Wolfgang Pauli came to know of Compton's discovery in the summer of 1923 when he came to Hamburg on several weeks'leave of absence from Copenhagen. He immediately got interested in the problem of thermal equilibrium between radiation and free electrons. Lorentz had reported in the first Solvay Conference that he could not establish the condition of thermal equilibrium for electrons having a Maxwellian velocity distribution [Lorentz's report in Langevin and de Broglie, 19121. Pauli wished to re-examine the problem by considering reversible collisions between electrons and directed light-quanta, obeying the laws of conservation of energy and momentum. Pauli [I9231 was able t o show that thermal equilibrium characterized by Planck's law was possible for electrons having a Maxwellian distribution provided the probability of collisions per unit time was given by

where A and B are the coefficients of Einstein's radiation theory [Einstein, 19171 and p, and p,, the densities of radiation before and after the collision respectively. Pauli showed that the first term was dominant for radiation satisfying Wien's law while the second term was more important for radiation of longer wavelengths satisfying the Rayleigh-Jeans law. Pauli interpreted this as the effect of interference fluctuations of classical waves. Einstein and Ehrenfest welcomed Pauli's work and clarified and generalized i t t o cases in which the elementary processes involved more than two light- quanta [Einstein and Ehrenfest, 19231. In particular, they discussed the 'paradoxical' second term in (23) 'which implies that the number of the elementary processes of scattering occurring per unit time a t an electron (which is a t quasi rest) increases faster than being proportional t o the radiation density p' of frequencies v', which the


radiation quantum possesses after being modified by the elementary process.' They concluded that this term followed from a detailed considertion of all interaction processes between radiation and electrons, including the induced emission of light (Einstein's 'negative radiation').

i. Bose's derivation While the leading European and American physicists were skeptical, even hostile to Einstein's Iight-quantum hypothesis, two unknown young physicists in India, M N Saha and S N Bose grasped its importance and started using i t with great success. Within two years of Einstein's 1917 paper from which he drew the revolutionary conclusion that a light-quantum carried directed momentum hv/c, M N Saha, a close friend of Bose, applied i t to explain the existence of radiation pressure on objects of molecular size [Saha, 19191. On the basis of the classical theory Nicholson [I9141 and Klotz [I9181 had shown that radiation pressure on particles of molecular size should become totally 'evanescent'. Saha applied 'quantum theory in the place of the old continuous theory of light. Instead of assuming that "lightn is spread continuously over all points of space,' he supposed that 'they are localized in pulses of energy hv . . . Let this pulse encounter a molecule m and be absorbed by it. Then in doing so the molecule will be thrust forward with an impulsive momentum of hv/c, (c=velocity of light); for we may suppose the pulse to have the mass hv/c2 and the momentum hvlc; the absorption of the pulse by the molecule may be taken as a case of inelastic impact, the whole momentum being communicated t o the molecule. The velocity with which the molecule will move forward = hvlmc.' Saha concluded that 'radiation-pressure may exert an effect on the atoms and molecules which are out of all proportion to their actual sizes.' This must have been the first application of Einstein's hypothesis that light- quanta carried a directed momentum hvl-c.

In March 1924 Saha visited Dhaka and stayed with Bose. While teaching postgradu- ate students Bose had keenly felt the need for a derivation of Planck's law free of logical difficulties. Saha drew Bose's attention t o the papers of Pauli [I9231 and their connection to Einstein and Ehrenfest [I9231 and their connection t o Einstein's paper of 1917. In an interview with J Mehra [Mehra and Rechenberg, 1,2,5651 Bose recalled : What seemed to be happening in Pauli's work was that in order t o apply the quantum conditions you had t o know exactly what was going to happen afterwards.' Stimulated by his discussions with Saha, Bose began to study carefully the worh of Planck [1900,19101, Peter Debye [1910,19231, Einstein [19171, Arthur Compton [19231, Pauli El9231 and Einstein and Ehrenfest [19231. This resulted in two papers that he wrote in June 1924 [Bose, 1924a, 1924bl and sent to Einstein for his views and for publication in Zeitschrift fiir Physik. Einstein translated both of them into German and had them published inzeitschrift fiir Physik in 1924. To both of them he added his own comments.

It is clear from these two papers of Bose which he always regarded as integral parts of each other that he clearly recognized the following points :

(1) The fundamental assumptions of the quantum theory are incompatible with the laws of classical electrodynamics [Bose, 1924al.

(2) All previous derivations had made use of relation (2) 'between the density of radiation and the average energy of an oscillator, and they make assumptions


about the number of degrees of freedom of ether, which enters tfie above equation (the first factor on the right hand side). This factor could however be derived only from the classical theory. This is an unsatisfactory feature in all derivations and i t is not surprising that efforts are made again and again to give a derivation free from this logical flaw.' [Bose, 1924al Even Einstein's 191611917 derivation, although 'remarkably elegant', makes use of Wien's displaqement law and Bohr's correspondence principle. Wi&~'s law is based on classical theory and the correspondence principle assumes that the quantum theory agrees with the classical theory in certain limiting cases.' [Bose, 1924a1. 'In all cases it appears to me,' wrote Bose 11924al 'that the derivations are not sufficiently justified from a logical point of view. On the other hand, the light-quantum hypothesis combined with statistical mechanics (as adapted by Planck to conform to the requirements of quantum theory) appears sufficient for the deduction of the law independent of classical theory' (our italics). In other words, the incompatibility between Einstein's light-quanta ('disjointed quanta') and Planck's law (implying 'undisjointed quanta') can be removed by adapting the statistical mechanics of light-quanta to conform to the requirements of quantum theory, as proposed by Planck. 'The problem of thermodynamic equilibrium of radiation in the presence of material particles can however be studied using the methods of statistical mechanics, independently of any special assumption about the mechanism of the elementary processes on which the energy exchange depends.' [Bose, J924bl

Bose therefore proceeded in four steps.

First step: He took Einstein's light-quantum hypothesis seriously and treated black- body radiation as a collectiop of light-quanta enclosed in a volume V. If there are N,

lightquanta of energy hv, ( s = 0 s = 0 1, the total energy E is

p, can be determinedofrom this simple relation if N, is known. So, the problem is to determine N,. 'If we can give the probability for each distribution characterized by arbitrary values of N,, then the solution is determined by the condition that this probability is a maximum subject to the subsidiary condition' (24) [Bose,1924al. Bose then proceeded to find this probability.

Second step: Before he could do that, he needed to compute the number of states of a light-quantum whose frequency lies between v, and v, + dv,. In order to be able to do this, he extended Planck's idea of dividing up the phase space of a material oscillator into elementary cells of volume h3 to the phase space of light-quanta. In other words, he extended Planck's 'first quantization' of material oscillators to the radiation field itself. This was the most novel and crucial input. Emboldened by Compton's discovery, he assumed the momentum of a light-quantum of frequency v to be hvlc in the direction of its motion. He must have also been aware of Saha's success in explaining radiation


pressure on molecular-size objects with the help of this hypothesis [Saha,19191. The instantaneous state of such a light-quantum, he stated, is characterized by a point in six-dimensional phase space which is constrained to lie on the surface of a cylinder defined by

Then the phase space belonging to the frequency interval dv, is

h dv, h3 V: dr dy dz dp, dpy clp, = 4 n V - = 4n - Vdv, .

C c3

Dividing this by h3, the 'ultimate elementary region' [Bose to Einstein, 4 June 19241, Bose obtained 4xV$ dv,/c3. Then the following statement occurs : 'In order to take into account the fact of polarization, i t seems however appropriate to multiply this number once again by 2' (our italics) to obtain 8nVv: dv,/c3 as the number of cells belonging t o dv,. No explanation is offered as to how this 'fact of polarization', an essentially classical concept, can be understood in terms of the light-quantum hypothesis, although Bose claimed to deduce this factor independent of the classical electrodynamics [Bose to Einstein, 4 June 19241. Bose had always maintained privately that he did offer a quantum theoretic explanation but Einstein removed it from his translation and substituted it by the statement about the polarization factor. In his letter t o Bose, Einstein simply wrote : 'You are the first person t o derive the factor quantum theoretically, even though because of the polarization factor 2, not wholly rigorously' [Einstein t o Bose, 2 July 19241. What was the explanation that Bose claimed t o have offered? It was that light-quanta carried an intrinsic spin that could take only the values k h/2n. There is only one recorded evidence of this because Bose's original manuscript in English is missing from the Einstein archives. I t appears in a paper by C V Raman and S Bhagavantam [19311. The paper is entitled 'Experimental Proof of the Photon Spin'. They write :

In his well-known derivation of the Planck radiation formula from quantum statistics, Prof. S.N. Bose obtained an expression for the number of cells in phase space occupied by the radiation, and found himself obliged t o multiply it by a numerical factor 2 in order to derive from it the correct number of possible arrangements of the quantum in unit volume. The paper as published did not contain a detailed discussion of the necessity for the introduction of this factor, but we understand from a personal communication by Prof. Bose that he envisaged the possibility of the quantum possessing besides energy hv and momentum hv/c also an intrinsic spin angular momentum Ht/2n round an axis parallel to the direction of its motion. The weight factor 2 thus arises from the possibility of the spin of the quantum


Bose's letter of 4 June 1924 to Einstein

S N Bose : The Man a d His Work

Einstein's postcard of 2 July 1924 to Bose


being right-handed or left-handed, corresponding to the two alternative signs of the angular momentum. There is a fundamental difference between this idea, and the well-known result of classical electrodynamics to which attention was drawn by Poynting and more fully developed by Abraham that a beam of light may in certain circumstances possess angular momentum . . . Thus, according to the classical field theory, the angular momentum associated with a quantum of energy is not uniquely defined, while according to the view we are concerned with in the present paper, the photon has always an angular momentum having a definite numerical value of a Bohr unit with one or other of the two possible alternative signs.

Why did Einstein remove Bose's explanation? Probably he felt it was too radical and unnecessary a t that stage. However, instead of removing it altogether, he could have added a suitable comment to the one he did make a t the end of the paper. In fact, he did not hesitate to add a dissenting note to Bose's second paper [I924131 as we shall see later.

Be that as i t may, let us proceed with Bose's derivation. Having divided the phase space volume belonging to the frequency range v, and v, + dv, by h8, he' wrote : 'the total number of cells must be regarded as the number of possible arrangements of a quantum in a given volume'(our italics). Classically there are infinitely many ways of arranging or distributing a particle within a phase space cell of finite extent corresponding to the infinitely many points contained within it. However, according to Planck, as we have seen, quantum states are not continuously distributed in phase space but are separated from one another by finite ainounts determined by h. Thus, an elementary cell of volume h3 can contain a t most one quantum state, and the problem of counting the number of quantum states reduces to the counting of these elementary cells. This is a totally new interpretation of the factor A, = 8x v: vdv,/cS. (In classical wave theory i t is the number of transverse standing modes in the interval dv, in a volume V.) In his covering letter to Einstein Bose wrote :

Respected Sit,

I have ventured to send you the accompanying article for your perusal and opinion. I a m anxious to know what you think. You will see that I have tried to deduce the coefficient in Planck's law independent of the classical electrodynamics, only assuming that the ultimate elementary regions in Phase-space have the content h3.

[Bose to Einstein, 4 June 19241.

On 12 July 1924 Einstein wrote to Ehrenfest : 'The Indian Bose has given a beautiful derivation of Planck's law, including the constant 1i.e. 8m2/cS I.'

Third Step: Having demonstrated that the factor 8xv2/cS in Planck's law corresponds to the total number of quantum states of radiation, Bose then proceeded 'to calculate the thermodynamic probability (macroscopically defined) of a state' in the sense of Planck (our italics). This is equal to the number of different microscopically defined states by means of which the macroscopic state can be realized. If all the cells (states) were considered to be distinct from one another, the answer would be A,! However, this


is not the case. Since each cell is to be counted as a single quantum state, if follows that the permutations of the light-quanta within a cell do not produce new states. Conse- quently, the cells can be partitioned into distinct classes characterized solely by their occupation numbers. Let there be a particular distribution ofN, quanta amongA, states such that there are p states that are empty, p states with one quantum, p 82 states with two quanta, and so on. Then, following Boltzmann and the usual procedure of statistical mechanics for these states (not the quanta), one can immediately write down the possible number of such distributions as

which is precisely the expression (12) given by Natanson. The 'thermodynamic probability' of 'the state defined by all p", is then clearly W = 7 W, which is equivalent to

the Debye-Planck expression (10). Fourth step : Bose then followed the standard procedure, already followed by Debye

[19101, of maximizing In W subject t o the constraints E = % N, hv, and N, = C rp>o

obtain Planck's law. Notice that the total number of photons N = C N, and there is no constraint on N. It turns out that the constraint on N, is not necessary, but Bose did not notice this.

Notice that the only departure of the expression (27) from the usual Boltzmann expression lies not in the product of factorials in the denominator but in putting the multiplicative factor (giY corresponding t o the possible arrangements of r quanta within each cell i havinggi levels equal to unity, as demanded by the division of phase space into elementary cells of volume h3. (In his second paper [1924bl written more or less a t the same time as the paper under discussion and received by Einstein only five days later, Bose used these factors, as we shall see later, for material particles obeying Maxwell-Boltzmann statistics.) This is precisely the adaptation of statistical mechanics by Planck 'to conform to the requirements ofthe quantum theory' that Bose referred to in the third paragraph of his paper.

Was Bose aware of the analysis of the Planck-Debye definition of the probability by Natanson and by Ehrenfest and Kamerlingh Onnes? One cannot tell. He certainly did not refer to their papers. I t is reasonable, however, to assume that he was not aware of the latter work because if he had been, he would have explained how their conclusion, namely that 'Planck's formal device (distribution of P energy elements E among N resonators) cannot be interpreted in the sense of Einstein's light-quanta' [Ehrenfest and Kamerlingh Onnes, 1914, p.8731, could be reconciled with his own starting point, namely the combination ofthe light-quantum hypothesis with Planck's counting device. The explanation, of course, lies in the fact that Planck's law reduces to Wien's law in the limit hv/kT > > 1, and that i t is only in this limit that the light-quanta have Maxwell-Boltzmann behaviour as inferred by Einstein. I t is one of those quirks of history that Einstein should have been able to infer the quantum nature of radiation from its approximate classical statistical behaviour in a certain domain and thus missed the true nature of its statistics. It was left to Bose to discover it, possibly


uninhibited by the analysis of Ehrenfest and Kamerlingh Onnes and their conclusion. Hence Einstein's enthusiastic welcome of Bose's work as an important and new support to his favourite light-quantum hypothesis, making it compatible with Planck's law.

j. The birth of a new statistics: Einstein applies Bose's method to ideal gases Bose's counting of states, of course, implied a new statistics for the light-quanta. Neither he nor Einstein seemed to have quite grasped the importance of this initially. In his letter of 2 July 1924 to Bose, Einstein compliments him for being the first to derive the factor A, quantum theoretically but does not write a word about the new statistics. 'In my opinion,'wrote Einstein in a footnote to Bose's paper, 'Bose's derivation signifies an important advance. The method used here gives the quantum theory of an ideal gas as I will work out elsewhere.' He extended Bose's method to material particles in three communications to the Prussian Academy in Berlin on 10 July 1924, 8 January 1925 and 29 January 1925 [Einstein, 1924,1925a, 1925131 without informing Bose or asking for his collaboration. Bose came to know of the comment in the footnote shortly before he left for Europe and of the first of the communications to the Prussian Academy on his arrival in Europe in October 1924. In the first paper this is how Einstein described Bose's method :

[First] : The phase space of an elementary object (here of a monatomic molecule and in Bose's case of a light-quantum), associated with a given (three-dimensional) volume, is divided into "cells" of extension ha. [Second] : If many elementary objects are present then their (microscopic) distribution, which plays a role in thermodynamics, is determined by the particular manner how the elementary objects are distributed among the cells. [Third] : The "probability" of a macroscopically defined state (in the sense of Planck) is equal to the number of different microscopic states by means of which the macroscopic state can be thought to be realized. [Fourth] : The entropy of the macroscopic state and, therefore, the statistical and thermodynamic behaviour of the system is determined by Boltzmann's theorem relating entropy and probability.' [Einstein, 1924, p.2611

Again, there is no mention of the new statistics! In applying Bose's method to ordinary material atoms, Einstein had to introduce an additional parameter (the chemical potential which vanishes for light-quanta) to take into account the conservation of their number. He drew a very important conclusion from this:

According to the theory presented here, Nernst's theorem is satisfied in the case of ideal gases. To be sure, our formulae cannot immediately be applied to extremely low temperatures, for we have assumed in their derivation that the pS, change only relatively infinitely little if s is altered by 1. Still one recognizes a t once that the entropy must vanish a t the absolute zero of temperature. The reason is that then all molecules are in the first cell; and for this state there exists only one distribution of molecules according to our counting method. Hence our assertion is immediately proved to be correct. [Einstein, 1924, p.2651 (our italics)

52 S N Bose : The Man and Ms Work

This is a conclusion of far-reaching consequence, yet there is no mention of any new statistics for the molecules; the emphasis is on Bose's counting method for states (one quantum state per cell). I t was only six months later on 8 January 1925 that Einstein discussed the difference between the counting methods of Boltzmann and Bose and recognized the fact, already pointed out by Ehrenfest [19241, that the new statistics for molecules was inconsistent with their statistical independence. However, while Planck [I9251 considered the analogy with radiation a weak point in Einstein's gas theory, the statistics of radiation and that of material particles being completely different in his opinion, Einstein regarded this result differently. He concluded : 'This result represents in itself a support of the view concerning the deep natural relation between radiation and gas, since the same statistical treatment which leads to Planck's formula establishes - when applied to ideal gases - the agreement with Nernst's theorem.' [Einstein, 1925a, p.71 (our italics) He was of the opinion that an understanding of the 'mysterious influence' among the molecules would come in the future.

k. The birth of wave mechanics Based on the new counting method, he showed in his second paper communicated on 8 January 1925 that the mean square energy fluctuation of the gas molecules is given by an expression which is the sum of two terms, one corresponding to the Maxwell- Boltzmann statistics of non-interacting molecules and the other to interference fluctuations associated with wave phenomena. (There exists in the Einstein Archives in the Jewish National and University Library, Jerusalem, a two page calculation done by Bose entitled 'Fluctuation in density' which ends with this fundamental .result. The document is undated and exists in the Scientific Correspondence File Folder 'B-Misc.- 11'. Bose was in Paris from 18 October 1924 until the first part of October 1925.) From this he immediately drew attention to the importance of Louis de Broglie's Doctoral thesis (which he had heard of from Paul Langevin and asked for a copy which he received and read in December 1924 [Jammer, 1966, 2491) in which de Broglie had attached wave properties to ponderable matter in analogy with the wave-particle duality of radiation. Einstein wrote : 'I shall discuss this interpretation in greater detail because I believe that it involves more than a mere analogy.' He then went on to explore various consequences of de Broglie's hypothesis.

When Erwin Schrodinger whose principal research during his early years in Zurich (1921-27) was on the statistical thermodynamics of ideal gases, read Einstein's first paper [1924al on the quantum theory of an ideal gas, he also did not understand that the molecules were being treated as being indistinguishable. On 5 February 1925 he wrote to Einstein suggesting a possible error in his probability formula. Einstein replied explaining that the 'quanta or molecules are not treated as independent of one another' and even gave a little diagram to explain the difference between Bose's counting method and that of Boltzmann. He concluded his letter with the emphatic statement: 'There is certainly no error in my calculation.' [Walter Moore, 1989,1831 It was Einstein's remark in his second paper that de Broglie's idea 'involves more than a mere analogy' that really induced Schrodinger to study the 'de Broglie-Einstein undulatory theory' as he called i t a t that time, and eventually led to his discovery of wave mechanics in 1926 [Schrodinger, 1926a, 1926b and 1926~1. Schrodinger later said that


Calculation of 'Fluctuation in density' in Bose's hand (Scientific Correspondence File Folder 'B-Misc.-11'); by courtesy of The Jewish National and University Library, Jeru- salem.



'wave mechanics was born in statistics.'(our italics) [Walter Moore, 1989, 1881 He also wrote to Einstein on 23 April 1926 : 'By the way, the whole thing would not have started a t present or a t any other time (I mean, as far as I am concerned) had not your second paper on the degenerate gas directed my attention to the importance of de Broglie's ideas.' [Jammer,1966,2571

1. Further developments Meanwhile in July 1925 Werner Heisenberg had sent a paper to Zeitschrift fiir Physik in which he gave a preliminary account of matrix mechanics [W Heisenberg'l9251. Schrodinger proved the complete equivalence of his wave mechanics and the Heisen- berg-Borndordan matrix mechanics [Schrodinger, 1926dl. A whole train of rapid developments occurred. Pauli discovered his Exclusion Principle within six months of Bose's paper [Pauli, 19251; Fermi published his paper on the statistics of particles obeying Pauli's principle in early 1926 [Fermi, 19261; Dirac linked the Bose and Fermi statistics of particles t o the symmetry properties of their wave functions and named them 'bosons' and 'fermions' [Dirac, 19261; in 1927 Dirac invented 'second quantization' of the electromagnetic field [Dirac, 19271 which was soon followed by a similar method for Fermi fields developed by Jordan and Wigner [19281; later Pauli [I9401 proved the spin-statistics theorem in relativistic quantum field theory; Pauli El9271 also applied Fermi-Dirac statistics to the paramagnetism of an electron gas and Sommerfeld [1927,19281 applied i t to the electrons in a metal; and Fritz London [I9381 suggested that the superfluidity of liquid helium ( * ~ e ) was related to Einstein-Bose condensation. Thus the developments of Schrodinger's wave mechanics and that of quantum statistical mechanics were directly triggered off by Bose's first paper followed by Einstein's far-sighted extension of his method to ideal gases of material particles and his championing of de Broglie's ideas which soon received experimental confirmation in the hands of Davisson and Germer [I9271 and G P Thomson C19271.

Years later this is how Schrodinger summed up the position : The two equivalent ways of looking a t (7.19) either as counting the number of quantum states of a particle, or as counting the number of wave-mechanical proper vibrations ofthe enclosure, interest us for this reason. The second attitude makes us think of the 'ns particles present in state as' as of a proper vibration (or a 'hohlraum' oscillator to use a customary phrase) in its n,th quantum level. (This attitude really corresponds to so;called second quantization or field quantization.) ns becomes a quantum number and the stipulation that the system of quantum numbers

determines only one state of the gas, not a class of

states, ceases to be a strange new adoption, and comes into line with the ordinary view about quantum states and their statistical weight (viz. equal


for any two of them).

It is the first, the particle attitude, that has suggested the term 'new statistics' which is frequently used. And that is why this idea of new statistics did not, originally, arise in comexion with heat radiation, because here the wave point of view was the historical one, the classical one-nobody thought of any other at the outset. The wave picture was considered to be (and historically was) the classical description. The quantizatian of the waves therefore .duly appeared t o be a 'first' quantization and nobody thought of anything like 'second quantization'.

Not until the idea of photons had gained considerable ground did Bose (about 1924) point out that we could, alternatively to the 'hohlraum' oscillator statistics, speak of photon statistics, but then we had t o make it 'Bose statistics'; Very soon after, Einstein applied the same to the particles of an ideal gas. And thereupon I pointed out that we could also in this case speak of ordinary statistics, applied t o the wave mechanical proper vibrations which correspond t o the motion of the particles of the gas [Schrodinger, 1952, 49-501.

We wish only to point out, as explained above, that Bose did not actually 'speak of photon statistics' -it was implied by his counting of quantized photon states, and this only became gradually clear with Einstein's application of Bose's counting method t o ideal gases, although Ehrenfest and Natanson had already pointed this out in 1911 and 1914 respectively in relation to Planck's counting method of December 1900 which Bose showed was applicable to light-quanta.

m. Did Bose make a 'shot in the dark'? Some recent widely read scientific biographies have given rise t o the impression that Bose made 'a shot in the dark' with three assumptions all of which turned out t o be correct but none of which he tried to justify 'by arguments of any kind; they seemed to appear intuitively to his mind, perhaps because his thoughts were uncluttered by the ongoing controversies of the European physicists.' [W. Moore, 1989,181; A. Pais, 1982, 4281 The assumptions were that photons were (a) massless particles capable of existing in two states of polarization, (b) indistinguishable and (c) their number was not necessarily conserved. The record must be put right. That photons are massless particles with energy hv and directed momentum hvlc was proposed by Einstein and only used by Bose. That they exist in two possible 'polarization states'was inferred by Bose from phase space considerations which yield for the number of quantum states between v, and v, + dv, the factor 4xv4 dv,/c3 rather than 8xvv: dv, /c3, the factor that occurs in Planck's formula. Actually, as we have already discussed, in order t o account for this factor of 2, Bose had proposed that photons carried one unit of intrinsic spin which could only take the values *I. Whoever had heard of particles with polarization? Moreover, 'polarization' was a wave concept which Bose wanted to avoid. However, Einstein quietly dropped it and the statement regarding the polarization factor was most probably inserted by him. Since Bose's original manuscript in English


is missing from the Einstein archives and Bose did not keep a copy, there is no way to verify this.

How about 'indistinguishability? Again, as we have seen, although Bose did not explicitly use the word 'indistinguishable' which is by itself rather va ue, he did use F Planck's division of phase space into elementary cells of volume h to justify his counting method for states (which implied indistinguishability of the quanta), namely that there cannot be more than one quantum state in an elementary cell. 'In any case,' he wrote, 'the total number of cells must be regarded as the number of possible arrangements of a quantum in the given volume.' What is not usually appreciated in usual expositions of Bose statistics is that Bose did not count the number of ways of distributing individual photons over a set of states. Instead, he calculated the thermodynamic probability of a macroscopic state by counting the number of ways in which the macroscopic state can be thought t g be realized through given distributions of the number of photons over the micro-states (phase space cells).

Finally, what about non-conservation of photons? This is a non-issue. Since, like Debye [19101, Bose considered the statistical thermodynamics of an open system of radiation in thermal equilibrium and did not include the molecules of matter with which this radiation constantly exchanged energy (which he did in his second paper which we shall turn to presently), there was no question of conservation of energy quanta even if treated as massless particles. In his 1917 derivation of Planck's law Einstein did not impose the conservation of light-quanta which he certainly regarded as massless particles interacting with molecules. Since Bose was not doing anything new or different in this respect, there was no original or deep comment he could have made on this issue. Nevertheless, i t is true that Bose, like most others who have made revolutionary contributions, did not, and could not be expected to, understand the full implications of their discoveries.

n. Bose's second paper (1924) We now turn to his second paper [I9241 on 'Thermal Equilibrium in Radiation Field in the Presence of Matter ', also translated into German probably by Einstein himself. This paper, completed on 14 June 1924, i.e. more or less a t the same time as the first paper, consists of two parts. In the first part Bose derived general conditions for statistical equilibrium of a system consisting of matter and radiation, independent of any special assumptions about the mechanism of the elementary radiative processes. In the second part he proposed a new expression for the probability of these elementary radiative processes which differed from that of Einstein. Einstein considered this hypothesis not to be 'applicable'to elementary radiative processes and gave two reasons for i t in a footnote to the paper. As a consequence, the paper has been largely ignored. However, Bose remained firmly of the opinion for the rest of his life that Einstein had not done him justice.

Bose started off by giving a quick critical review of the most important derivations of Planck's law until then given by Debye [19101, Einstein [19171, Pauli [19231, and Einstein and Ehrenfest [I9231 on the scattering of radiation by electrons. Bose considered Debye's 1910 derivation not to be 'completely independent of classical electrodynamics' since he derived the factor 8xv: Vdv,/cS from the normal modes of the ether.


He considered the derivations of Einstein [19171, Pauli [I9231 and Einstein and Ehrenfest [I9231 to be dependent on ad hoc special assumptions about the probability of elementary radiative processes. 'The problem of thermodynamic equilibrium of radiation in the presence of material particles,' he wrote, 'can however be studied using the methods of statistical mechanics, independently of any special assumption about the mechanism of the elementary processes on which the energy exchange depends. In this way we obtain a general relation which is valid for all special assumptions about the elementary processes and their probabilities.' This is in keeping with Kirchhoff's law. He then went on to say: 'If i t is possible to write down the thermodynamic probability for any particular distribution of quanta of radiation and the probability for any arbitrary energy distribution among the particles, then the thermodynamic probability for the bigger system consisting of matter and radiation is simply the product of the two probabilities. The condition of equilibrium is simply that the resulting probability must be a maximum.' For the thermodynamic probability for radiation, Bose used the expression

(A, + N, dv) ! w = n , A, ! N, dv !

where A, = 8nvv2 dv/c3 which, he stated, had been derived earlier. As references he quoted his own paper and that of Debye [Debye 19101. (It is curious that the reference to his previous paper is given as 'The author, to appear in Phil Mag'. This is because Bose had submitted his paper first to the Philosophical Magazine. Not hearing from them for a while, he sent a copy to Einstein. While translating Bose's second paper, the translator obviously forgot to change the reference of the earlier paper to Zeitschrifl far Physik 26, 178-181, 1924). Although expression (28) is indeed the Ansatz used by Debye, it does not appear in Bose's first paper! Instead, the corresponding expression that appears in his first paper is (271, the form given by Natanson. As we have seen earlier, the two are mathematically equivalent, but Bose did not care to explain this, probably because this result must have been fairly well known in the literature in those days.

For the thermodynamic probability for material particles. Bose writes:

This can be found easily. We want to generalize our assumptions a little so that the case of the Bohr atom with discrete energy levels as well as the case of translational energy of particles can be included. Let the phase space be divided into cells. For every cell there is a probability g that a particle occupies it. The g's are in general equal except for the case of Bohr's atoms. The thermodynamic probability for any arbitrary distribution nl, nz etc among the different cells is,


(our italics). Unlike in the case of radiation (expression (271, Bose now uses the factor ll g:i for material particles and obtains the classical Maxwell-Boltzmann distribution. i

In other words, he treats material particles as in classical (distinguishable) statistical mechanics but light-quanta according to quantum (indistinguishable) statistical mechanics.

Finally, he writes down the thermodynamic probability for the total system as the product of (28) and (29) :

with

and

XNT = N.

He then considers an elementary process in which a particle passes from the rth cell to the sth cell while a light-quantum of frequency v changes into a light-quantum of frequency v.' The stationarity of W subject to the conditions (31) and (32) gives

where

x ~ V ' - Z ~ V + E , - E , = 0 .

Bose then demonstrated that (33) indeed generalized the results of Pauli [I9231 and Einstein and Ehrenfest [I9231 without the need for any ad hoc assumptions regarding the elementary radiative processes designed to give Planck's formula. In particular, in the case of Bohr's atoms considered by Einstein in 1917, equation (33) reduces to

which is Einstein's condition derived on the assumption that atoms in higher energy levels make transitions to lower energy levels in two ways:

1. Aspontaneous transition as in radioactivity independent of the state of the external radiation field.

2. An induced transition whose probability depends on the state of the external radiation field.


Transitions from lower to higher energy levels always take place by induced absorption whose probability depends on the state of the external radiation field. Further, Einstein had to postulate certain relations (namely, the opposite limiting Wien and Rayleigh- Jeans laws) among these transition probabilities to obtain Planck's law. Bose's derivation is independent of all these assumptions. I t is for thisreason that Einstein took the trouble to have the paper translated into German and communicated i t to Zeit- schrift fiir Physik. However, his disagreement with Bose started from the next part of the paper in which Bose proposed a new expression for the probability of an interaction between a particle and a quantum of radiation.

Bose's starting point was the following obse=vation which he considered to be fundamental :

. . . even in a collision no interaction is as probable as the occurrence of any special interaction. . . From the classical theory. . . one would expect that there is some interaction whenever a quantum and a material particle come together. Therefore here it is a question of a departure from classical theory.

The assumption made above, that even in a collision no interaction need occur, is quite analogous to the assumption of the stability of stationary states which is so fundamental to Bohr's theory of line emission and can be traced back to the same' origin-the probability of existence of the stationary states of the particles. I t is interesting to remark in this connec: tion that in the analogous case of a collision of an electron with an atom, experiments show that the electron goes through the atom without changing either the interatomic or its own motion. (our italics)

Bose was obviously referring to the Rarnsauer-Townsend effect. He saw a simple way of realizing the non-classical possibility in Planck's cellularization of phase space. Let p, be the number of cells with r quanta. 'In order that the radiation and particle may interact, it must be in a cell occupied by a quantum. The particular cell which the molecule under consideration occupies will pass through all possible states if we observe it long enough, that is, sometimees i t is empty, sometimes occupied by 1 quantum, sometimes by 2 quanta etc. The duration of these events will be finally proportional to the equilibrium values of p,, pl etc.When r quanta and a particle are together, then either an exchange of energy takes place or nothing happens.' (our italics) Therefore r + 1 different events are possible, namely the exchange of 1 quantum of energy, 2 quanta etc or no exchange a t all. Therefore the total number of possible events is

The number of cases in which interaction or energy exchange occurs is


Consequently the probability of an interaction is

This is Bose's second fundamental result. Having derived this result, Bose then considered Einstein's 1917 problem, namely

Bohr atoms exchanging energy with the radiation field. In order to be consistent with the fundamental equilibrium condition (351, he found i t was sufficient to consider (a) transitions from lower to higher states through absorption of quanta of characteristic frequency v and (b) transitions from higher to lower states to be spontaneous, i.e. independent of the external radiation field. The additional assumption of induced emission (negative radiation) processes was not necessary. Let the strength of induced absorption be p. Then the probabillity of induced absorption is

Let the coefficient of spontaneous emission be a. Then for equilibrium one gets

which agrees with (35) provided g, P = g, or. A substitution of the value of nJn, then gives Planck's law. Bose then treated the cases considered by Pauli [I9231 and Einstein and Ehrenfest [I9231 and showed that the fundamental equilibrium condition (33) followed from the probability law (38) without requiring any further assumptions.

In a note added to the paper Einstein gave two reasons why he thought Bose's hypothesis about the probability of elementary radiative processes was not 'applicable'. In his letter of 3 November 1924 Einstein summarized the two reasons. Your principle is not compatible with the following two conditions: 1) The absorption coefficient is independent of the radiation density. 2) The behaviour of a resonator in a radiation field should follow from the statistical

laws as a limiting case.' [Einstein to Bose, 3 November 19241 Let us see to what extent these criticisms were fair. As far as the first point is

concerned Einstein was referring to Beer's law. Bose's probability law, however, predicted a dependence of the absorption coefficient on the radiation density, decreasing with it. 'If the behaviour had been such,' wrote Einstein in the note, 'then i t would certainly have already been discovered in the case of infra-red radiation from hot light sources.' This is hard to believe. The departure from classical behaviour predicted by Bose's principle should occur only when Nv / A cc 1, i.e. for very low intensity radiation. I t is now well known that the departures from classical behaviour predicted by quantum optics are not seen even when the intensity of light is extremely low, unless the source happens to be of a special kind (like a single-photon source). The


reason is that light from all classical sources is in a coherent state in which the number fluctuation is extremely large (Poissonian or higher). Therefore only light from 'single- photon states' (Fock states) and 'squeezed states' and other such 'sub-Poissonian situations' show departures from classical behaviour. [Aspect, 1987; Loudon, 19831 Such light sources were certainly unknown in 1924.

The second point concerns the 'correspondence principle' (not necessarily Bohr's) which in this case implies that a radiation field can transfer positive as well as negative energy to a resonator, depending on the phase. 'The probabilities of both these transitions must depend on the density of radiation, that is, on N,, as opposed to Bose's hypothesis,'wrote Einstein. The discovery of masers and lasers after the Second World War eventually vindicated Einstein's contention. I t was nevertheless eventually an empirical rather than a sound theoretical refutation. Although the 'correspondence principle' was used as a heuristic tool with great success by Einstein, Planck and later the Copenhagen school prior to the discovery of quantum mechanics around 192511926, it is no longer regarded as a fundamental principle. In fact, quantum mechanics itself is known not to satisfy this principle. [Liboff, 1980; Cabrera and Kiwi, 1987; Bohm and Hiley, 19851 For example, the nodes of the eigen-solutions inside a box do not disappear for arbitrarily high quantum numbers so that a classical particle cannot pass through such points. Also the cross-section for hard sphere scattering has no classical limit. Einstein was aware of this and used i t against quantum mechanics as well.

o. Bose's third (unpublished) paper (1925) Although Bose was unhappy, particularly with the first point ragarding Beer's law, he accepted the second point, modified his stance and wrote a third paper which he sent to Einstein from Paris on 27 January 1925. He sent him a letter under separate cover in which he wrote :

It seems that the hypothesis of negative Einstrahlungstands, which, as you have yourself expressed, reflects the classical behaviour of a resonator in a fluctuating field. But the additional hypothesis of a spontaneous change, independent of the state of the field, seems to me not necessary. . . I am rather anxious to know your opinion about i t . I have shown it to Professor Langevin here and he seems to think it interesting and worth publishing.

It was never published. Neither did Bose keep a copy of the paper, nor is there one in the Einstein Archives as far as one can ascertain. So, the precise contents and details will remain unknown. Bose arrived in Berlin in October 1925 and discussed his new ideas with Einstein. Einstein was so convinced of the necessity and reality of the two independent processes of emission of light from atoms (spontaneous and induced) that he could not agree with Bose that one of them, the former, was unnecessary to assume in deriving Planck's law. To Bose, the process of emission of light appeared as a single physical process and the division into two independent processes appeared artificial. The assumption of one, he claimed, would automatically imply the other in thermal equilibrium, given the special statistical property of the radiation field. Initially Bose started with spontaneous emission as input but later changed his position. It was clear to Bose that Planck's law was a consequence of the special statistical property of the radiation field itself and was independent of the mechanisms of energy transfer with


Bose's letter of 27 January 1925 to Einstein



atoms and molecules. This was indeed his capital contribution to physics and forms the substance of his first paper and the first part of his second paper. To Einstein, on the other hand, it appeared that Planck's law was a consequence of special mechanisms of energy transfer between the radiation field and molecules and that thermal equilibrium was impossible without both induced and spontaneous emission. This was indeed the lesson he derived from his 191611917 papers which he considered to contain 'the fundamental outline of the future theoretical derivation.' Einstein was compelled to regard spontaneous emission as being independent of the state of the radiation field and an inherent property of atoms and molecules because in 1917 he was unaware of the special quantum statistical property of the radiation field. I t is strange that even aRer championing Bose's method he failed to see Bose's point about spontaneous emission. To Bose i t was clear that it depended on the 'environment'in which the atom or molecule was placed. He wrote in his second paper 119241 :

From classical theory one would expect that no stationary states are possible and that an interaction or coupling (Bohr) between ether and the excited atom and the radiation connected with i t is always present. On the other hand, in order to explain spectral emission we must assume i t to be possible that no interaction (emission) occurs. Because of the existence of stationary states we are therefore led to assign to every stationary state a probability coefficient or a mean life-time.

- Recall the expression (38) for the fundamental probability Z? I t can be written as n / ( Z + 1 ) where E = N, / A, is the average number of quanta per cell. For E > > 1, P tends to unity, the classical limit in which stationary states are not possible. I t follows from this that the average life-time of a stationary state (and therefore the probability of spontaneous emission) depends on the average radiation density E. Bose used to recount to his students Einstein's objection to this. 'Suppose there were only one hydrogen atom in the universe in an excited state,' Einstein once remarked to Bose, 'don't you think it would radiate spontaneously and come down to its ground state? Bose did not disagree, but nevertheless, felt that such a gedanken situation did not capture the situation in a Hohlraum or cavity. I t was Bose's intuition that eventually turned out to be right in the end, not Einstein's. Firstly, in Dirac's method of 'second quantization' (Dirac, 1927) which is widely used in quantum field theory today, one starts with a classical electromagnetic field in which the equality of the probabilities of induced emission and absorption (determined by the Fburier coefficients in the plane wave expansion) is built in, and obtains the spontaneous emission term as a consequence of the fundamental commutation rules. Although this is not exactly what Bose had proposed, i t fits in with his basic contention that spontaneous emission is a consequence of the quanta1 property of the radiation field itself and need not be introduced as an independent hypothesis concerning the radiating atoms. Einstein's initial reaction to Dirac's contribution was also decidedly negative. [Pais, 1982,4411

Secondly, in quantum electrodynamics spontaneous emission is not a property of an isolated atom but of an atom-vacuum system and can be significantly inhibited or enhanced by placing the atoms in a suitable environment. In quantum electrodynamics the vacuum is not empty and devoid of everything-it is full of virtual particles


appearing and disappearing, interacting with one another; i t provides a non-trivial physical 'environment' to an atom. A whole new branch of quantum optics called 'cavity quantum electrodynamics' has developed since about 1987 utilizing dramatic changes in spontaneous emission rates to construct new kinds of microscopic masers that operate with a single atom and a few photons or with photons emitted in pairs in a two-photon transition. This is what two of the pioneers in the field have to say :

Ever since Einstein demonstrated that spontaneous emission must occur if matter and radiation are to achieve thermal equilibrium, physicists have generally believed that excited atoms inevitably radiate. Spontaneous emission is so fundamental that it is usually regarded as an inherent property of matter. This view, however, overlooks the fact that spontaneous emission is not a property of an isolated atom but of an atom-va.cuum system. The most distinctive feature of such emission, irreversibility, comes about because an infinity of vacuum states is available to the radiated photon. If these states are modified - for instance, by placing the excited atom between mirrors or in a cavity- spontaneous emission can be greatly inhibited or enhanced.

Recently developed atomic and optical techniques have made i t possible to control and manipulate spontaneous emission. [Haroche and Kleppner, 19891

In his third unpublished paper Bose had gone further. He wrote in his letter of 27 January 1925 to Einstein :

I have tried to bok at the radiation field from a new standpoint and have sought to separate the propagation of Quantum of energy from the propagation of electro-magnetic influence. I seem to feel vaguely that some such separation is necessary if Quantum theory is to be brought in line with Generalized Relativity theory.

The views about the radiation field, which I have ventured to put forward, seem to be very much like what Bohr has recently expressed in May Phil Mag 1924. But it is only a guess, as Icannot say honestly to have exactly understood all he means to say about virtual fields and virtual oscillators.

This letter shows that Bose was one of the first persons to have formulated the idea of an 'empty wave'(an electromagnetic wave propagating in space and time but carrying no energy-momentum). He was certainly the first person to have intuitively seen its relationship with the principle of General Relativity, although the connection is not clear. Einstein already had the idea of an empty wave in his light-quantum hypothesis according to which 'a ray of light expands starting from a point, the energy does not distribute on ever increasing volumes, but remains constituted of a finite number of energy quanta localized in space and moving without subdividjng, and unable to be absorbed or emitted partially.' [Einstein, 19051

If the localized quanta carry all the energy and momentum, what happens to the electromagnetic waves which can produce interference? There was no clear-cut answer. The problem was so acute that Einstein referred to these waves as Gespensterfelder (ghost waves) guiding the photons. [Bohr, 1945,2061 Louis de Broglie also had a similar


idea in his concept of phase or pilot waves 'guiding the propagation of the energy' and enabling a 'synthesis of the waves and the quanta.' [de Broglie, 1923, 5491 Slater also arrived a t a similar notion. As he recalled : 'A number of scientists - W [illiam] F [rancisl G [ray] Swann, among others -had suggested that the purpose of the electric field was not to carry a continuously distributed density of energy, but to guide the photons in some manner. This was the point of view which appealed to me, and during my visit a t the Cavendish Laboratory in the fall of 1923, I elaborated on it. [Slater, 1975,91 Slater who believed in the existence of light-quanta as well as electromagnetic waves found the following way out of the difficulty : 'I decided you could only have a statistical connection between them because you couldn't set up a vector to represent the photon in electromagnetic theory : that you would have the intensity of the wave governing the probability of finding the photon there. And I wanted to have the wave emitted d&ing the stationary state so as to get i t emitted over a long enough period so that i t would have a suitable spectral distribution.' [Slater, loc.cit., 301

When Slater arrived in Copenhagen on 21 December 1923 to collaborate with Hendrik Kramers and Niels Bohr, he was prevailed upon by them to relinquish Einstein's light-quantum hypothesis but to retain his idea of electromagnetic waves emitted by oscillators during their stationary states and carrying no energy. They coined the name virtual oscillators for them. The collaboration resulted in the famous Bohr-KramersSlater proposal [Bohr, Kramers, Slater, 19241 which Bose referred to in his letter to Einstein. In the BKS proposal the conservation of energy (and momentum) was abandoned for individual transition processes in atoms but was retained as a statistical concept. Most interestingly, they wrote : 'the transitions which in [the Einstein theory of 19171 are designated as spontaneous are, in our view, induced by the virtual field.' Bose must have noted this with satisfaction and this must have been one reason why he regarded the BKS proposal to be 'very much like'his own. The theory was however quickly falsified by experiments carried out by Bothe and Geiger [Bothe and Geiger, 19251 and Compton and Simon [Compton and Simon, 1925a, bl which established the conservation of energy and momentum in individual processes. Never- theless, their ideas were reformulated and generalized by Born [I9261 in his well known statistical interpretation of the wave function in quantum mechanics, an idea now universally accepted. This is what Heisenberg wrote about its historical antecedent :

'The probability wave of Bohr, Kramers and Slater. . . was a quantitative version of the old concept "potentian in Aristotelian philosophy. I t introduced something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality

Later, when the mathematical framework of quantum theory was fixed, Born took up this idea of the probability wave and gave a clear definition of the mathematical quantity in the formalism.' [Heisenberg, 1958, 291

On 16 July 1946 , twenty-one years after he received Bose's letter of 27 January 1925 from Paris, Einstein wrote to Schrijdinger about the latest status of the unified field theory (of electromagnetism and gravitation based on the General Principle of Relativity):

'It is indeed correct that the energetics occurs quite differently here compared to Maxwell-Poynting. I consider, however, that in this respect the Maxwell theory is really


false on account of quantum actualities. In light the energy exists in something like quasi-singularities. The wave field as such should therefore not be the site of energy. One must simply require this and be happy that in the new theory the transverse wave field is indeed present but as such transports no energy.'(Our italics) [Moore, 1989,4271

There is no mention of Bose, his letter or his third paper which Einstein must have rejected outright, although the facsimile of Bose's letter supplied by The Jewish National and University Library, Jerusalem, shows that he had marked the phrase 'h.ave sought to separate the propagation of Quantum of energy from the propagation of electro-magnetic influence' within parentheses.

Louis de Broglie's original interpretation .of Schrodinger's equation [de Broglie, 19271 was given a firmer basis by Bohm 119523. In the de Broglie-Bohm 'casual interpretation' particles are thought t o exist independent of observation, embedded in a wave field y which satisfies the Schrodinger equation, with a density (over the

2 statistical ensemble) of value I yI . One can show that this approach can explain the interference pattern observed in double-slit experiments without abandoning the notion of well-defined particle trajectories. [Philippidis, Dewdney and Hiley, 19791 Experiments for testing certain consequences- of 'empty waves' have been proposed .[Selleri, 1990, ch. 41 and some have already been carried out [Zou, Grayson, Wang and Mandel, 19921. However, such experiments have not yet ruled out all versions of'empty wave' theories [Vigier and Holland, 19911.

Bose remained ever grateful to Einstein (whom he always addressed as 'Sir' or 'Master') for his encouragement and the interest he had taken in his first paper. Nevertheless, deep within himself, he felt a resentment that rankled all through his life. Three of his most perceptive and original ideas were scotched by Einstein :

1. The idea of the photon spin which is now established beyond doubt. (That Bose proposed the idea in his first paper in 1924 cannot as yet be established beyond every shred of doubt, although circumstantial evidences are str'ong.)

2. The idea that spontaneous emission is not an inherent property of an isolated atom, totally independent of the radiation field, also confirmed by experiments since 1987.

3. The idea of the propagation of electromagnetic influence carrying no quanta of energy, an area of current theoretical and experimental activity.

Bose could not publish a single paper during his two years in Europe a t a time when physics was going through a turmoil and one of its most exciting periods. He returned to Dhaka a disheartened man and did not publish anything in theoretical physics for the next twelve years !

The author is grateful to N Mukunda, V Singh and J C Pati for a critical reading of the manuscript and suggesting improvements.

PARTHA GHOSE S N Bose National Centre for Basic Sciences, Calcutta

Base Statistics : a historical perspective

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1925a Abhandlung, Sitz. ber. Preuss. Akad. Wiss. (Berlin), 3-14 (presented at the meeting of 8 January 1925). 1925b Sitz. ber. Preuss. Akad. Wiss. (Berlin), 18-25 (presented at the meeting of 29 January 1925). Einmtein, A and Ehredest, P. 1923 2. Phys. 19,301-306. Fermi, Enrico 1926 Rend. R. Accad. Lincei 3 145-149 (presented at the meeting of. 7 February 1926,Z. Phys. 36,902-912. Friedrich, W., Knipping , P. and Laue, Max von 1912 Sitz.ber. Bayer. Akad. Wiss. (Miinchen), 303-322. Haroche, S. and Kleppner, D. 1989 'Cavity Quantum Electrodynamics', Physics Tiday 42, 1,24-30. Heisenberg, Werner 1925'2. Phys. 33, 879-893; English translation in Sources of Quantum Mechanics (Van der Waerden, 1967), 261-276. 1958 Physics and Philosophy, The Revolution in Modern Science, Harmondsworth : Penguin Books, 1990. Jammer, Max 1966 Conceptual Development of Quantum Mechanics, New York, St. Louis, San Francisco, Toronto, London, Sydney : McGraw Hill Book Co. Jordan, P. and Wigner, E. P. 1928 2. Phys. 47,631. Klotz 1918 Jour. of the R.A.S. of Carncr&, 12,357. Langevin, P. and Broglie, Maurice de 1912 La Thkorie du Rayonnement et les Quanta -Rapports et Discussions de la Rkunion tenue cf Bruxelles, du 30 Octobre au 3 Novembre 1911, Paris : Gauthier-Villars. Liboff, R.L. 1987 Phys. lbday 87,50. London, Fritz 1938 Nature 141,643-644; Phys. Rev. 54,947-954. Loudon, R. 1983 The Quantum Theory of Light, 2nd ed. Clarendon, Oxford. Mehra, J. and Rechenberg, H. 1982 The ~ i s to r ica l Development of Quantum Theory, 1,l-2, New York, Heidelberg, Berlin : Springer Verlag. Millikan, R A 1916 Phys. Rev. (2) 7,355-388. Moore, Walter 1989 Schrodinger : Life and Thoughts, Cambridge, New York, Port Chester, Melbourne, Sydney : Cambridge University Press. Natanmon, Ladimlam 1911 Phys. 2s. 12, 659-666. Nicholson, S. B. 1914 Monthly Notices of the Royal Astron. Soc, 74,425. Pair, Abraham 1982 'Subtle is the Lord. . .' The science and life ofAlbert Einstein, Oxford, New York, Toronto, Melbourne : Oxford University Press. Pauli, Wolfgang

Bose Statistics : n historical perspective

1923 Z . Phys. 18,272-286. 1925 2. Phys. 31, 765-783. 1927 Z. Phys. 41, 81-102. 1940 Phys. Rev. 58, 716-722. Philippidie, C., Dewdney, C. and Hiley, B.J. 1979 Nuovo Cimento 52B, 15. Planck, Max 1900 Verh. d. Deutsch. Phys. Ges. ( 2 ) 2, 237-245. 1910 'Zur Theorie der Warmestrahlung., Ann. d. Phys. (4) 31, 758-768. 1911 'La loi du rayonnement noir e t I'hypothhse des quantites elementaires d'action' in Langevin and de Broglie, 1912, 93-114. 1925 Sitz. ber. Preuss. Akad. Wiss. (Berlin), 49-57. Raman, C. V. and Bhagavantam, S. 1931 Ind. J. Phys. 6,353-366. Saha, M S . 1919 Astrophys. Jour. 50,220. SchrBdinger, Erwin 1926a Ann. d. Phys ( 4 ) 79,361-376. 1926b Ann. d. Phys ( 4 ) 79,489-527. 1926c Ann. d. Phys (4) 80,437-490. 1926dAnn. d. Phys (4) 79, 734-756. 1952 Statistical Mechanics, A course of Seminar Lectures, London : Cambridge University Press, Second Edition. . Selleri, F. 1990 Quantum Paradoxes and Physical Reality, Dordrecht, Boston, London : Kluwer Academ- ic Publishers. Sommerfeld, A. 1927Atti del Congresso Internazionale dei Fisici. 11-20 Septembre, Bologna : Nicola Zanichelli (19281, 2 , 449-473; 1928 Z. Phys. 47,43-60; Ber. dt. chem. Ges. 61, 1171-1180. Thompson, G.P. 1928 Proc. Roy. Soc. A 117, 600. Vigier, J.P. and Holland, P. 1991 Phys. Rev. Lett. 67, 402. Zou, X.Y., Grayson, T., Wang, L.J. and Mandel, L. 1992 Phys. Rev. Lett. 68, 3667-3669.

S N Bose : Collected scientific

papers

On the Influence ofthe Finite k lume of Molecules on the Equation of ~ t a t r f ~ y MEGH NAD SAHA, M. SC., and SATYENDRA NATH BASU, M. SC., Lecturers on Mathematical Physics, Calcutta University.

I t i s wellknown that t i e departure of the actual behaviour of gases from the ideal NK6 state defined by the equati0n.p = - isdue to two causes : (1) the finiteness of the

v volume of the molecules, (2) the influence of the forces of cohesion, i.e. the attractive forces amongst the molecules. van dei Waals was the first to deduce an equation of state in which all these factors are taken into account; according to van der Waale, we have

where b = 4x volume of the molecules, a defines the forces of cohesion.

In all subsequent'modifications of this equation (Clausius, Dieterici, or D. Berthelot) the changes which have been proposed all relate to the influence of the cohesive forces; the part of the irgument dealing with the finiteness of molecular volumes is generally left untouched.

But it has been found that the results of experiments do not agree with the predictions of theory if we, regard a and b as absolute constants. Accordingly it has been proposed to regard both a and b as functions of volume and temperature.*

But before proceeding to these considerations, it is necessary fo scrutinize whether the influence of finite molecular volumes is properly represented by the term b. From theoretioal considerations, the conclusion has been reached that this is not the case. The wgument is as follows :

According to Boltzmann's theory,

the entropy S = K log W+C,

where K = Boltzmrtnn's gas-constant, W = probability of the state. Let us now calculate the probability that a number N of molecules originally confined within the volume Vo and possessing finite volumes, shall be contained in a volume. V. Neglecting

V the influence of internal forces, the probability for the first molecule is - ,for thesecond vo

v-P rriolecule the probability is - , where /? = 8 x volume of a single molecule, for when vo-B

t Communicated by the Authors. *Compare van der Waals, Proc. Amat. 1916 ; Van Lear, Proc. Amet. Vol. xvi, p. 44.


the first molecule is in position, the space enclosed by a ooncentrio sphere of double the radius of the molecule will not be available for the seoond moleoule. The avail-

able space is therefore V-/?, whence the probability is - '-' Introduoing similar 70-B '

mnsiderations for the rest of the molecules, we have

We are, of course, neglecting those cases in which partial overlapping of the regions occupied by two or more molecules occurs; for the number of such cases a n at best be a small fraction of the total number. Even cases of aotual association do not include these, for in that case, two discrete molecules become merged into one, without their outer surfaces being actually in contaot.

From the relations S = K log W+C

we can easily verify that

RB = -- V-26 26 log 7 ( R = N K ) ... (3)

NK8 Ae a first approximation, when 6 is small compared to v, we obtain p = - v ,(Boyle-Charles-Avogab h w ) , and as a, second approximation we obtain

NKO p =- (van der Waale oorrection). V-6

We also note that x pV = NKB.- where x = & 1-eJ: KO . .. (4)

To account for the infiuence of internal forces, we multiply, following the lead of a

Dieterici, the above expression (3) by e-~- having the same signifioance as. before. From this equation of state, we can ea& verify the following resulta for the

critical point :

2c Critical volume, V, = - b = 3.1686, e- 1

NKB K = - = 3.613. 7,

Finite Volume of Molecules on Equation of State 77

The corresponding values of V , from the van der Waals and the Dieterici equations 8 e2

are (3b, 2b) respectively, and of K are (x = 2.66, - 2 3.696 ) respectively.

As a matter of fact, for the simpler gases, the value of 'K ' obtaiued in this paper e2

agrees better with the experimental results that the Dieterici value we have for

oxygen* K = 3.346, for nitrogen? K = 3.53, for argon3 K = 3.424, for xenon** 8 .

K= 3.605. We need not consider the van-der-Waals value -, for it fails entirely. 3

The most serious drawback to Dieterioi's equation is, according to Prof. Lewis (vide Lewis's Physical Chemistry, vol. ii. p. 117) that it makes b or the limiting volume

5 3 bhile the limiting volun~e, obtained by the extrapolation of Cailletet-Mathias

mean density line to the temperature 0 = O°K is about - The value of b obtained 4 '

v c in this paper, viz., - therefore agrees better with this value. 3 16

It is yet premature to predict what influence this investigation will have on the speculations concerning the variability of the volume of nlolecules with temperature. A more detailed investigation dwelling upon this point, and the application of the fornlula (4) to Amagat's (pv, p) curves, will be communicated shortly. Meanwhile

a -- we point out that the factor e NKev has been introduced into the expression for

'p' only as a ,provisional measure, though it is considered that this step, thoughnot quite exact, is one in the right direction. In the next paper an &tempt will be made to introduce energy into probability calculations. Sir T. N. Palit Laboratory of Science,

Calcutta. Note added in proof-On consulting the literature on the subject, we noticed that in several papers in the Amsterdam Proceedings (vids vol xv, p. 240et seq.), Dr. Keesom of Leyddn had also made attempts to deduce the equation of state from Boltzmann's entropy principle. But, in the expression (2) for W , he introduces, before differentia-

b tion, an approximation in which terms up to second order in -are retained only. v

In this way, he arrives at the van der Waals' form v-b for the influence of finite molecular volumes. In obtaining our present equation of state (4), no such approximation has been made. (M. N. SAHA and S. N. BASU.)

*Mathias and I(. O n n e s , Proo. Amst. Feb. 1911. TBerthelot, Bull. de b Soo. France cle Phye. 167 (1901) $Methim, O n n e s , and Crommelin, Proo. Amst. 1913, p. 960, Vol. xv. **Paterson, CdRRS. Whvthw-Cfrav. Proa. Bov. Soc. Lond. A. kuxvi. D. 579 (1912). Reprinted from Phil Mag Ser 6,36, pp. 199-203,1918 (Taylor and Frands, London).

The S tress-Equations of Equilibrium

(Read April 6th, 1919)

It was shown by Mitchell that the six stress-coefficients in an isotropic medium satisfy six equations of the type :

Them equations however h v e not been used for solving the general problems of Elasticity. It is shown here, that the equations can be successfully integrated in the case of a semi-infinite body bounded by a plane. In the case of the sphere the equ- tions can be conveniently transformed, in a different form, which then admit of integration in an infinite series of spherical harmonics.

(1) The semi-injnite solid bounded by z = 0

The surface tractions X, Y, Zz, are supposed to haw given values over the plane z = 0

Consider the equations

Since 0 is a harmonic function the' general solution can be wt.itfen as

The Stress-Equations of Equilibrium

where X,,., Y,,, Z,,, are harmonic funotions vyhich have given values X,Y,Z,, over the plane z = 0.

The functions me therefore uniquely determined; they are in fact:

1 a X, I a Xzo = - - J J - dxdy, Yz0 = -- 2n dz r 2n az J J- > dxdy.

1 d 2, Z - - J J - dxdy, 'O - 2n dz r

also since

ax, a ~ , az, -+ -+--- = 0 ax ay az

we have from (1)

where

Thus X,, Y,, Z,, 0 are d l determined.

The solution may be afterwards completed, and U, V, W found out as in Cerrutti's method.

( 2 ) The problem of the wphere. Consider the three equations

80 S N Bose : The M a n and His Work

Multiplying by x, y, z, andadding we have

Now since

we have

8 0 where r -- - 0 is also a harmonic function. dr

The form of the equations is exactly similar to the preceding equations.

(3) The sphere of radiw a, has given tractions X,, Y, , Z,, over the surfuce.

Assuming

where Q, is a solid homogenoous harmonic of the nth degree,

a 0 r-- 0 = Z(x-1 )0 , ar

also remembering that

v8(ra-a2)Pn = 2(2n+3)PS, where P, is a solid homogeneous h o - nic of the nth degree, we clee the solutions of the equation (2) can be expreeeed in the form

The Stress-Equations of Equilibrium

1 n- 1 r z , = - - a o n C - (r2-a2)-+a&, 2 ( l+a ) 2n+l dz

where X,, Y,,, Z,,, are harmonic functions which have given values over the aurfaoe of the sphere T = a and hence are completely determined. If

X, = Z.Xn7 Y, = Z Y,, I;,, = I: 2,;

we have

Again

ax, ax, ax, ar, a ~ , a ~ , = ~ [ ~ + F + ~ l + ~ I ~ + ~ - + ~ l az, az, +z [=+ ay +$ ] +x,+ y,+zz = 0,

it follows that

82

so that

S N Boae : The Man and His Work

where

So that

Thus X,, Y,, Z,, 0 are all determined in terms of the known value of X,, Y,, Zt, on the surface.

Reprinted from Bull Cal Math Soc 10, pp. 117-121, 1919 (Calcutta Mathematical Society).

On the Herpolhode

(Read August 31st, 1919)

X . de Sparre has shown that the horpolhode of Poinsot contains no point of inflexion; his proof was based upon the properties of elliptic functions. Various other proofs of the theorem have since been given by Mannheim, Saint-Germaina, Routh3, Le- Cornu* and others.

The following simple proof of the theorem is based solely on the dynamical considerations of the Poinsot motion; hence it might appear interesting and instructive.

Let A , B, C be the three principal moments of inertia of a body fixed at a point 0; let 01 and 01' be the two hstantaneous axes, at time t and t+dt, I, I t , etc., an the invariable plane trace out the horpolhode, length of 01, 01', etc. being proportional to the resulting angular velocity at all times.

At a point of inflexion of the horpolhode, there will be a stationary tangent, i.e., the total change of the vector II' will be then along its own length.

Remarking that 11' has components proportional to w1 dt, w, dt, w, dt, along the three moving principal axes, the condition reduces to

w1 wa We have from Euler's equation8

1An exactly similar proof of the theorem discuesed in this paper h*s been given by Prof. W. Van der Woude in a paper eptitled de Herpoloide Van Poinsot", publish4 in Nieuw Archief voor Wis- kunde, Tweede Reeks, Dee1 XII, Eerste Stuk, pp. 94, 95 (1919). It appears however from an enquiry that this journal was not received in the library of the Society before the beginning of November, 1919 -8. K. B.

2 Compte Rendu-1885 SAdvanced Dynamics. 'Bulletin de 800. Math. de France-1906.

84

from which


. . Awl = ( B - C ) ( ~ ~ ~ + & W , )

6 .

Bw, = ( c - A ) ( ~ & u , + ~ , w ~ )

CW, = (A-B)(wl wt+WawI) Substituting in (1) we have

and two similar equations.

So that

But remembering that

Aw, wl+ B W , ~ , + C W ~ W , = 0

we have

hence substituting, we get as a necessary condition for the point of infiexion

which is obviously impossible because B+C- A, C+ A- B, A+ B-C, are all positive, so that the left-hand side is an essentially positive quantity.

Reprinted fromBt~ll Ca1 Math Soc 11. pp. 21-22, 1919 (Calcutta Mathematical Society).

ON THE EQUATION OF STATE.

To the Editors of the Philosophical Magazine. Sirs, -

In a paper published in the 'Philosophical Magazine' for August 1918 it has been shown that from Boltzmann's theory of entropy we can m i v e at an equation of state

Amongst other applications of this theorem it was shown that the value of the criti-

ROC 2e 8 cal constant K = - would be ---- =3.53, instead of van der Waals' -and Dieterici's PC Ve e-1 3

A table was given showing that in the case of the most of the elementary gases S

the value 3.53 corresponded better with experimental results than either-- or 3.69. 3

The list comprised He, N,, O,, Xe, but not Hydrogen, about which the experimental data were not satisfactory.

Recently the critical data for Hydrogen have been re-determined in the Laboratory of Prof. K. Onn& (vide Proc. K. Akad. Wetenschappen. Amsterdam, Vol. xx. 191 7).

I t has been found that

8, = 33".18K, d, = .0310, p, = 12.8 atmospheres;

from these data, K = 3.35.

It is superfluous to add that this value of K is in much better agreement with the 8 ea

value 3.53 than with either the value- = 2.66 or - = 3.69, thus corroborating the 3 2 belief expressed in the aforesaid paper that the equation (I), though not final, marks a step in the right direction.

University College of Science, Calcutta, India.

Reprinted from Phil Mag Ser 6,39, p. 456,1920 (Taylor and Francis, London).

On the Deduction of Rydberg's Law from the Quantum Theory of Spectral Emission." By SATYENDRA NATH BASU, M. SC., University Lecturer in Physics, University College of Science, Calcutta.

It is well known that Rutherford's model of the atom has been fruitful in explaining many facts connected with atomic radiation. In the simplest case of hydrogen, with a nucleus consisting of a single positive charge, and an electron, Dr. Bohr has successfully applied the quantum theory to explain the Balmer series of hydrogen spectra. The mathematical problem of finding the spectral series for any atomic system has since been clearly formulated by Sommerfeld, and the quanta condition has been generalized in a form suitable for system with any number of degrees of freedom. If q,, q,, q,, ... q, are co-ordinates to fix the position of the electron responsible for emission, and pl, pp, p3, . . . , p n are the corresponding generalized momenta, any statical path, according to Sommerfeld, is characterized by the conditions

where n's are whole numbers and h is Planck's constant, the integral being extended generally over the complete orbit. The radiation is supposed to take place when the electron jumps from one statical path to another. The difference in energy, a t the same time, flows away in the. form of a homogeneous radiation of frequency v,

which can be calculated from the Bohr equation hv = W1- W,. Sommerfeld hw successfully applied this conception in explaining the fine structure of hydrogen lines. It is clear, however, that the problem of theoretically calculating the spectrum of any atom other than hydrogen is beset with difficulties of a formidable nature. It is exactly analogous to the dynamicd problem of "n" bodies, where only in favourable cases we are able to find solutions. Nevertheless, from a purely experimental standpoint, we know that the visible radiation from any element can be classified in definite series. The frequency of any line in the series c m be expressed as a difference of two terms, each of which has the form

where m is a whole number and a and /3 are two constants depending upon the element and the nature of the series. So that if we are to explain the formation of the series

- -

* Communicated by the Author. +Bohr, Phil. Meg. July 1913. **Sommerfeld, Ann. der Physik, li (1916).

On the Deduction of Rydberg's Law 87

from theoretical considerations following Bohr and Sommerfeld we must look upon each member multiplied by "h" as giving the energy of the atomic system when the radiating electron woves in a definite statical path. The complexity. of the inner atomic field under which the radiating electron moves .is to be lookedupon as bringing in the terms involving a and b. So A seems interesting to see what will be the corr- responding expression for energy in a system by which the oomplex nature of the internal field may be approxi&tely represented. In the case of any atom we have, in general, a condensed nuclear charge of +ne (where n'is the atomic number) surrounded by rings of electron at different distances. The number of eleotrons in total must be also equal to n in order to secure that the atom is electrically neutral in the ordinary state.

In X-ray emission the electron displaced comes from the inner rings; in the case of visible radiation, however, we have reasons to think that the displaced eleotron responsible for radiation comes from the outermost ring-the valenoy electrons, as they have been designated by Sommerfeld. When excited for radiation, we can suppose that the electron in the outermost ring is removed to a greater distance from the centre than the others, so that the force acting upon it may be regarded as the resultant of the various forces exerted by the central charge and the remaining electrons. The potential at any point can be regarded as given by

where r is the distance from the centre and r, is the distance from the s-th electron. If we neglect the influence of the moving electron upon the arrangement of the others surrounding the nucleus, it is clear that the potential can be approximately represented

ea as --+ L*. The resultant field might be looked upon as due to a single positive r r2 charge together with a doublet of strength L in a certain fixed direction which we take as our Z-axis. If we neglect the disturbing effect of the outer electron, L may be taken to be approximately fixed in direction and in magnitude in the small interval of time during which the active emission takes place.

We may, therefore, take as our model a system consisting of a positive oharge and a doublet of strength L. We proceed to calculate the energy in a statical path on the above simplified hypothesis.

The kinetic energy of the moving electron is obviously

the potential energy


ea eL cos 6 v = --+ r ra '

Two integrals can be at once written down :

mra sin' 86 = c,,

2ea 2 h cos 8 m[&++&+ra mn a @] - - + r4

= -W. r

To get another integral, we write

d eLsin6 - (mr48)-mra sin 6 cos 8& = dt ra '

Integrated, it gives

The expressions for three impulses mr, mr4 and mra sinWq5 osn now be written down : in terms of the oomtants of integration we have

mra sin Wq5 = c1

The quanta conditions can be written down. a,s

o r 2 sina i$ dg5 = nlh,

m r 3 d6 = nzh,

I m&r = n,h,

- W being twice the total energy of the system. The integrals are to be extended over the whole range within which the expression within the square root remains positive.

On the Deduction of Rydberg's Law

The integration.

From (1) we have obviously

of the two remaining expressions, (3) can be integrated most easily : in fact,

gives after integration

The second integral can be written as

by putting cos 0 = x.

The right-hand side is to be integrated throughout the region, when the cubio remains positive. It cannot be integrated in finite terms; an approximation suitable for our purpose can, however, be made, assuming 2meL to be small compared with (c2-cf) = A. To see what this means we are to remember that c2-c; is of the dimension of h2; so that 2 m L must be small compared with ha, or L must be small compared

ha with -. Now, if a , -P,.and y are taken as the three roots of the cubic, the cubic

me

where y is the greatest of the positive roots and D = 2meL. The limits of the integral are obviously a and -p,

Hence

l i - xdx m = f -8 ~ D + z ) ( u - x ) ( x + ~ ) ] ~ ~

90

Supposing


a - x = (a+p) oosa 8,

p+x = sipa 8,

we get

where 1 and K are the usual elliptio integrals, defined by

a12

On the sssumption that D is small, we have

So that, expanding E end K and making necleasery approximations, we have finally

Also

On the Deduction o f Rydberg's Law

So that we have

Now, collecting all the quanta conditions, we have

So that we have

and

Assuming

we have

or

Calling

we have

where

we have approximately


So that

So that the energy in the statical path

if we suppose ,that the spectrum is due to ionized a t o m in which the field can be approximately represented as a central charge of E = pe and a doublet of different strength L'.

We have, by a similh reasoning, energy in a statical path

where N = Rydberg number.

As a result of numerous investigations on the nature of the spectral series, i t has been shown that for many elements the different series can be grouped according to the following schemes : P-Series :

v = (1.5, 4-(m, p1) 1 m = 2, 3, 4... -(m, P')

11. Subordinate Series : v = (2, p1)-(m+'5,8)

nz = 2, 3, 4... V = (2, p 2 ) - ( ~ + ' s , S) .

1st Subordinake Series : v = (2, pl)-(m, d).

Companion : v = (2, p1)--(m, d')

On the Deduction of Rydberg's Law

N Symbol (m, f ) stands for according to Rydberg, and

N (m+f )" ( m + f + - q 2

m2

according to Ritz.

The frequency of the lines emitted thus appears as the difference of two terms, each of which is to be regarded as corresponding to the energy in a definite statical path, on the Quantum theory. Sommerfeld* has recently given reasons for assuming that in any series of statical orbits corresponding to the different kinds of s, p, and d terms the azimuthal quqnta generally preserve a certain definite value, whereas the radial quanta can have all values from 0 to co; he thus shows that in s, p and d terms the azimuthal quanta generally have values 1, 2, and 3 respectively.

Making the above assumptions in our formula, we see that the expression of the

energy comes out as - Nh in the same form as required by the Ryd-

(n,+n,+n,+A)2 berg formula. The constants A, however, depends only upon n, and n,; it dimini- shes for increasing values of the azimuthal quanta; so that they decrease progressively in the s, p, arid d terms. Moreover, our form shows that A depends upon n, m d n, separately, so that for the same value of nl+n2 we may have different value8 of the constant. Thus, if we suppose n,+n2 = 1 we have two values corresponding to the vaues 1, 0 and 0, 1; for n,+n2 = 2 we have three values; and so on. Thus we see, even on Somnierfeld's assumption, for the constancy of the azimuthal quanta we shall hive two different s, three different p, four different d terms. At least two different values of p and three different values of d seem to be required by the series formula, which is essential for the explanation of doublets and triplets of constant frequency difference**. We thus see that our model serves a t least as a qualitative explanation of the following facts :

(1) The progressive decrease of the characteristic numbers in the 8 , p, and d terms,

(2) The existence of different sets of s, p, and d t e r m for the same element. It is clear, however, that our simplified assumption will not fit in any actual case

exactly. The complex nature of the internal field can in no case be properly repre-

sented by a simple term, --- Oos ', in the Potential. Moreover, we have reason to believe r2

that the internal arrangement of the electrons itself will be influenced, in a large measure, by the motion of the outer electron, which we have neglected in our formula. In fact, Land6-f has tried in a recent paper to take account of this disturbance in the

* Sommerfeld, Verh. d. Phys. Ges. May 1919. ** If we exclude the case n2 = 0-Le., if we assume that the motion in s plane containing the

axk of the doublet is excluded-we get the proper number of 8, p , d terms as observed in the case of the alkali metals and the doublet system of alkaline earths.

Land& Phys. Zeit. 1909.


comparative simple case of the helium series. But at the same time, it is hoped that the calculation, in this comparativgly simple case, will serve to illustrate a t least some general principles a t which we have arrived by an experimental study of the spectral series.

In this paper an attempt has been made to deduce the laws of regularity in the spectral series of elements on the basis of Bohr's qukntum theory of spectral emission. Starting from Sommerfeld's assumption that the ordinary line-spectra of elements are due to the vibration of one outer electron (the valency electron), i t has been shown that the field of the nucleus and the remaining (n-1) electrons may be represented

eZ eL cos 6 . by the Potential V =- - + - -- i.e., the field due to a single charge plus a r r 2

doublet of strength L. The axis of the doublet is variable, but the emission is supposed to take place so quickly that in that short time the axis does not appreciably change.

The qumta conditions have been applied according to Sommerfeld's rule,

nh = I fi da , and the energy of the system has been reduced to the quanta numbers. The energy comes out in the form

in the paper, where n, is the radial quantum, n is the azimuthal quantum, and z is given by an equation of the sixth degree, involving only the azimuthal quantum, and is a function of n only.

It has been next shown that if, in accordance with Sommerfeld's prinoiple, we assume n = 1 for the 8-orbits, n = 2 for the p-orbits, n = 3 for the d-orbits, n = 4 for the b-orbits, then, with a very simple assumption, we obtain a single value for the energy of the s orbits, a double value for the energy in the p-orbit, a treble value for the d-orbit. Then, applying Bob's law hv = W,- W,,, we arrive at Rydberg's laws of the regularity in spectral series, in the case of the alkali metals.

Exact calculations are not tried on account of the uncertainty of the value of L ; but it has been pointed out that the values of 8 , (p,, p,), (dl, d,, d,) progressively decrease, as is actually the owe.

If the value of L be supposed to vary with n, the radial quantum, then probably the above calculations would lead to Ritz's law.

Reprinted from Phil Mag 40, pp. 619-627, 1920 (Taylor and Francis, London).

Planc k s Gesetz und Lichtquantenhypothese. Von Base (Dacca-University, Indien).

(Eingegangen am 2. Juli 1924.)

Der Phasenraum eines Lichtquants in bezug a d ein gegebenes Volumen wird in .Zellenu von der GroDe h3 aufgeteilt. Die Zahl der moglichen Verteilungen der Lichtquanten einer makroskopisch definierten Strahlong nnter diese Zellen liefert die Entropie nnd damit alle thermodynamischen Eigenschaften der Strahlung.

P l a n c k s Formel fiir die Verteilung der Energie in der Strahlung des schwarzen Korpers bildet den Susgangspunlit fur die Quantentheorie, welche in den letzten 20 Jahren entwickelt worden ist und in d e n Gebieten der Physilr reiche Friichte getragen hat. Seit der PubUation im Jahre 1901 sind viele Arten der Ableitung dieses Gesetzes vorgeschlagen worden. Es ist anerkannt, d d die fundamentalen T70raus- setzungen der Quantentheorie unvereinbar sind mit den Gesetzen der klassischen Elektrodynamilr. U e bisherigen Ableitungen machen Ge- brauch von der Relation

d. h. von der Relation zwischen der Strahlungsdichte und der rnittleren Energie eines Oszillators, und sie machen ,Amahmen iiber die Zahl der Freiheitsgrade des &hers, wie sie in obige Gleichung eingeht (erster Faktor der rechten Seite). Dieser Faktor konnte jedoch nur aus der klassischen Theorie hergeleitet werden. Dies ist der unbefriedigende Punkt in d e n Ableitungen, und es kann nicht wundernehmen, d d immer wieder Anstren,o;ungen gemacht werden, eine Ableitung zu geben, die von diesem logischen Fehler frei ist.

Eine bemerkenswert elegante Ableitung ist von E ins t e in angegeben worden. Dieser hat den logischen Mangel d e r bisherigen Ableitungen erkannt und versucht, die Formel unabhiingig von der klassischen Theorie zu deduzieren. \Ton sehr einfachen h a h m e n uber den Energieaustausch zwischen Molekiilen und Strahlungsfeld ausgehend, findet er die Relation


Indessen m d er, urn diese Formel mit der Planckschen in merein- stimmung zu bringen, von W i ens Verschie-bungsgesetz nnd B o h r s Korre- spondenzprinzip Gebrauch machen. W i en s Gesetz ist auf die klassische

Theorie gegriindet, und das Korrespondenzprinzip nimmt an, dad die Quantentheorie snit der klassischen Theorie in gewissen GrenzfUen iibereinstimme.

In d e n F U e n scheinen mir die Ableihgen nicht geniigend logisch gerechtfertigt. Dagegen scheint mir die Lichtquantenhypothese in Ver- bindung mit der statistischen Mechanik (wie sie durch P lanck den Bediirfnissen der Quantentheorie angepdt worden ist) fiir die Ableitang des Gesetzes unabhirsgig von der klassischen Theorie hinreiehend zu sein. Im folgenden will ich die Methode kurz skizzieren.'

Die Strahlung sei in das Volumen T eingeschlossen nnd ihre Ge- samtenergie E sei gegeben. Es seien verschiedene Sorten von Quasten von der jeweiligen h a h l N, und Energie hv, vorhanden (s = 0 bis s = oc). Die Toklenergie E ist dann

E = ~ i V , h v , = ~ / ~ ~ d v . a

(1)

Die Lasnng des Problems verlangt dann die Bestiinmung der N,, welche g, bestimmen. Wenn wir die Wahrscheinlichkeit fiir jede dnrch be- liebige N, chasakterisierte Verteilung w e b e n kdnnen, dann wird die Lasmg durch die Bedingung bestimmt, daS diese Wahrscheinlichkeit bei wahrung der Febenbedingung (1) ein Sdaximnm sein soll. Diese Wahr- scheinlichkeit wollen wir nun adsuchen.

h v , Das Quantum hat ein Xoment vom Betrage - in der Richtung C

seiner Fortbewegang. Der Yomentanzustand des Quantums wird cha- rakterisiert durch seine Koordinaten 1;, y, ; and die zugeharigen Yomente p, p, p, ; diese sechs GroDen kannen d s Pnnktkoordinaten in einem sechs- dimensionalen Raum anfgefdt werden, wobei wir die Beziehung haben

vermoge welcher der genannte Punkt auf einer durch die Frequenz des Quants bestimmten Zylinderfiiiche zu bleiben gezmungen ist. Z w

Plancks Gesetz und Lichtquantenhypothese

Frequenzbereich dv , gehijrt in diesem Sinne der Phasenraum

Wenn wir das gesamte Phasenvolumen in Zellen von der GroDe h3 ein- v2

teilen, gehoren zum. Frequenzbereich d v also 4 n P - d v Zellen. In c3

bezug auf die Art dieser Einteilung kann nichts Bestimmtes gesagt werden.

Indessen muD 'die Gesamtzahl der Zellen als die Zahl der moglichen An- ordnungen eines Quants in dem gegebenen- Volumen angesehen werden. Um der Tatsache der Polarisation Rechnung zn tragen, erscheint es dagegen geboten, diese Zahl noch mit 2 zu multiplizieren, so daO wir fiir

v2 dv die Zahl der zn dv gehijrigen Zellen 8 z V - erhalten.

c3 h'nn ist es einfach, die thermodynamische Wahrscheinlichkeit eines

(makroskopisch definierten) Zustandes zu berechnen. Es sei iIT8 die Zahl der zum Frequenzbereich d fl gehorigen Quanten. Auf wie riele Arten konnen diese a d die zu d fl gehorigen Zellen verteilt werden ? Sei yp, die Zahl der vakanten Zellen, p; die Zahl derer, die ein Quant enthalten, pi die Zahl der Zellen, die zwei Quanten enthalten usf. Die Zahl der moglichen Verteilungen ist d a m

A 8 1 8 nu2 , wobei A 8 = - p t ! p;! . . . c3 d +,

nnd wobei N8 = 0 .p; + 1 .p8, + 2 p ; . . .

die Zahl der zu dv8 gehorigen Quanten ist. Die Wahrscheinlichkeit des durch ssmtliche p: definierten Zustandes

ist offenbar

8

a t Riicksicht darauf, daO wir die ps+ als groOe Zahlen betrachten konnen, haben wir


wobei

Dieser Ausdruck soll ein Maximum sein unter der h'ebenbedin,oung

E = x ~ 8 h v ' ; N8 = x r P : . 8 f

Die Durchfiihrung der Variation liefert die Bedingungen

Hieraus f olgt *

Daraus folgt zun&hsC rhvd --

g = B a e P . Da aber

rhvd -- d 8 = x ~ * e P = l ? * ( 1 - e 9

r so ist

( -7). B8 = A8 1 - e Ferner hat man

rh vb N 8 = x r p : = x r A 8 ( l - - e

r 8 r

Nit Rucksicht a d den oben gefnndenen Wert von A* ist also

Plancks Gesetz und Lichtquantenhypothese

Xit Benutzung der bisherigen Resultate findet man ferner

d s 1 woraus mit Riicksicht 'darauf, dad - = - folgt, daD = kT. Setzt

dE T' man dies in obige Gleichung fiir E ein, so erhalt man

welche Gleichung P 1 an c k s Formel iiquivalent ist. (fiersetzt von A. E ins te in . )

h n m e r k u n g des Ubersetzers . Boses Ableitung der Planck- schen Formel bedeutet nach meiner Meinung einen wichtigen Fortschritt. Die hier benutzte Methode liefert auch die. Quantentheorie des idealen Gases, wie ich an anderer Stelle ausfiihren will.

Reprinted from Z Physik 26, pp. 168-171, 1924 (Springer - Verlag, Heidelberg).

Planck's Law and the Light-Quantum Hypothesis

Bose (Dacca-University, India) (Received 2 July, 1924)

The phase space of a light-quantum in a given volume is divided up in 'cells' of size h3. The number of possible distributions of the light-quanta of a macroscopically defined radiation among the cells gives the entropy and with that all the thermodynamic properties of the radiation.

Planck's formula for the distribution of energy in the radiation of a blackbody forms the starting point for the quantum theory which has been developed in the last 20 years and has been very fruitful in all parts of physics. Since its publication in 1901 many methods for the derivation of this law has been proposed. I t is recognized that the fundamental assumptions of the quantum theory are incompatible with the laws of classical electrodynamics All derivations till now use the relation

that is, the relation between the density of radiation and the average energy of an oscillator, and they make assumptions about the number of degrees of freedom of the ether, which enters the above equation (the first factor on the right hand side). This factor could, however, be derived,only from the classical theory. This is an unsatisfactory feature in all derivations, and it is not surprising that efforts are made again and again to give a derivation free from this logical flaw.

Einstein has given a remarkably elegant derivation. He has recognized the logical flaw in all previous derivations and has tried t o deduce the formula independently of classical' theory. Starting from very simple assumptions about the energy exchange between molecules and the radiation field, he finds the relation

In order to make this formula agree with that of Planck he has to use Wien's displacement law and Bohr's correspondence principle. Wien's law is- based on classical theory and the correspondence principle assumes that the quantum theory agrees with the classical theory in certain limiting cases.

In all cases i t appears t o me that the derivations are not sufficiently justified from a logical point of view. On the other hand, the light-quantum hypothesis combined with statistical mechanics (as adapted by Planck t o cornform to the requirements of quan-

Planck's Law and the Light-quantum Hypothesis 101

tum theory) appears sufficient for the deduction of the law independent of classical theory. In the following I shall sketch the method briefly.

Let the radiation be enclosed in a volume V and its total energy be E. Suppose there are different types of quanta each having number N, and energy hv, (s = o to s = w 1. The total energy is then

r

The.solution of the problem then requires the determination of of Ns which in turn determine p, . If we can give the probability for each distribution characterized by arbitrary values of N,, then the solution is determined by the condition that this probability is a maximum subject t o the subsidiary condition (1). We now want to find this probability.

hvs The quantum has a momentum of magnitude - in the direction of its motion. The C

instantaneous state of the quantum is characterised by its coordinates x, y, z and the corresponding momenta p, p, p,. These six quantities can be considered to be the coordinates of a point in a six dimensional space, where we have the relation

by virtue of which the point is forced to lie on the surface of a cylinder determined by the frequency of the quantum. The phase space belonging to the frequency interval dvb is.

v2 If we divide the total phase space volume in cells of size h3, then 4 R V y dv cells will C

belong to the frequency interval dv. Nothing definite can be said about the method of this division. In any case, the total number of cells must be regarded as the number of possible arrangements of a quantum in the given volume. I t seems, however, appropriate to multiply this number once again by 2 in order to take into account the fact of

v2 dv polarization, so that we obtain 8 R V - as the number of cells belonging to dv.

c3 Now i t is easy to calculate the thermodynamic probability (macroscopically defined)

of a state. Let N" be the number of quanta belonging to the frequency range dvS . In how many ways can these be distributed among the cells belonging to dvS ? Let pi be


the number of empty cells,p",hose which contain one quantum, pS, those which contain two quanta and so on. The number of possible distributions is then,

A 8 ! 8 z v a , where A8 = - pe,! p?! . . . c9

d f l ,

and NU = 0 .pb, + 1 . p ; + 2 p ; . . .

is the number of quanta belong to the range dvS . The probability of the state defined by allps is clearly

Taking into account that we can considerp; to be large numbers, we have

~ g w = ~ 8 1 ~ ~8 - xp;~gp; , where 8

8 r

This expression must be a maximum with the constraints

8 r

Carrying out the variation, we obtain the conditions

From this it follows that

Next i t follows from this tha t

Planck's Law and the Light-quantum Hypothesis

But since -- rh*r A # = X B ~ ~ P = * ( I - a ? ) - ' ,

r

therefore

We further have

Taking into account the value of A' found above, we get

Using the results found so far, one further finds

as 1 Whence, using the relation - = - if follows that P = k T . Substituting this in the a E T'

equation for E, we get

which is the same as Planck's formula.

(Translated by A. Einstein)

Translator's remarks : In my opinion Bose's derivation signifies an important advance. The method used

here gives the quantum theory of an ideal gas as I will work out elsewhere.

English translation of Z Phyeik 86, pp. 168-171,1924 (Springer - Verlag, Heidelberg).

Wiirmegleichgewicht im Strahlungsfeld bei Anwesenheit von Materie.

Von S. N. Bose in Ramna (Indien).

Die Wahrscheinlichkeit eines beliebigen Zustendes eines aus Strahlung und Molekeln bestehenden Systems wird berechnet, und es werden hiereus die Bedingungen fur des statistische Gleichgawicht abgeleitet. Im Anschlup hieran werden neue Auadrucke fiir die statistische Wahrscheinlichkeit der Ebmen- tarvorghge vorgeschlagen, welche der Wechselwirkung zwischen Strahlung and Materie entsprechen.

Debyel) hat gezeigt, dab das Plandksohe Gesetz mit W e der statistischen Mechanik abgeleitet werden kann. Seine Ableitung ist jedoch ineofern nicht vijllig unbhangig von der klassischen Elektrodynamik, als er Gebrauoh macht von dem

Begriff der Eigensohwingungen des ~ t h e r s und annimmt, da/3 hinsichtlioh der Energie

8.AV8 V dv Reso- das Spektralgebiet zwischen v und v+dv ersetzt werden kann durch - cs

natoren, deren Energie nur Vielfache von hv betragen kann. Man kann aber zeigen, dab die Ableitung so abgelndert werden karin, dap man iiberhaupt keine Anleihen

8nvZ bei der klassischen Theorie zu machen braucht. -

c3 V dv k m gedeutet werden

als die Anzahl Elementargebiete des schesdimensionalen Phasenraumes fiir die Quanten. Die weitere Rechnung bleibt im wesentlishen dieselbe.

Einstein hat sich einer anderen Methode bedient. Er. betrachtet die Wechsel- wirkung zwischen den materiellen Teliohen und dem Strahlungsfeld. Kennt man die Energieverteilung untr den materiellen Teilchen, so kann man das Gesetz der aohwarzen Strahlung ermitteln aus der Bedingung der Stationaritiit dieser Verteilung beim Aust.ausch von Energie zwisohen Materie und Strahlung. Dieser Austausch sit weiterhin eine Folge gewisser Elementqrozesse. Eine gwignete Formulierung der Eigenschaften und Wahrscheinlichkeit des Elementar prozesses ermoglicht es, daa Plancksche Gesetz abzuleiten, wenn man die Energieverteilung unter den Teilchen als gegeben ansieht. In seiner ersten Arbeit8 sind dib materiellen Gebilde Bohrsche Atome, die nur einer diskreten Reihe stationrirer Zustiinde fahig sind. Energieaus- tausch findet statt infolge von Emissionsund Absroptionsprozessen und gleichzeitigsn

uberg&ngen der Atome von einem Zustand zu einem anderen. Er hat dargetan, dab das Plmoksohe Gesetz sich ergibt, wenn den Emissions- und Absorptionsprozessen gewisse Wahrscheinlichkeiten zugeschrieben werden. Neuerdings hat aber das I'roblem

(1) Debye, Ann. d. Phys. 33, 1427, 1910. (2) Einstein, Phya. ZS. 18, 121. 1917.

Warmegleichgewicht im Strahlungsfeld. 105

des thermischen Gleichgewichts zwischen dem Strahlungsfeld und freien Elektronen Wichtigkeit erlamgst durch die Arbeiten von Debye3) Compton4) und anderen, die sich mit der Zerstreuung der Strahlung durch Elektronen befassen, Pauli5) hat dieses Problem in einer interessanten Arbeit diskutiert. Als Elementarprozep betraohtet er die Streunng, durch die ein Quant, das in einer bestimmten Richtung fliegt und innerhalb eines bestimmten Spektralintervalls,liegt, sich in ein anderes eines mderen Spektralinter~~lls und von verschiedener R.ichtung verwandelt. Gleichzeitig nndert ein Elektron, das sich mit einer gewissen Geschwindigkeit in einer bestimnlten Richtung bewegt Grope und Richtung seiner Geschwindigkeit. Fur alle diese Pro- zesse gilt das Erhdtiingsgesetz von Impuls u d Energie. Pauli hat gezeigt, dab, wenn die RTnhrscheinlichkeit dieses Elementarprozesses (Ap+Bppl)dt ist, wo p und p' die Strahlungsdichten der Frequenzen v und v', A und B unabhangig von q and q' aber abhanging von der Art des betrachteten Stopes sind, dieses Wahrscheilichkeits- gesetz zum Planckschen Gesetz fiihrt, falls fur das Elektronengas das Maxwellsche Verteilungsgesetz giiltig ist. Einstein unde Ehrenfest haben das Paulische Ergebnis noch verallge~~leinert(~). Sie betrachten einen Prozep, bei dem ein Molekul order ein Elektron Qumten hv,, hv2 usw. absorbiert und gleichzeitig hv;, hvk . . .emittiert. Es wird angenommen, dab die emittierten und absorbierten Quanten bestimmte R'ichtungcn Ilaben. Zugleich andert sich auch Grope und Richtung der Geschwin- digkeit des betreffenden Molekiils. Energie und Impuls bleiben aber erhaken. Das Wahrscheinlichkeitsgesetz von Pauli wird folgenderma /? on verallgemeinert :

fur den direkten und inverson Prozep.

Er zeigt, dap Gleichgewicht vorhanden ist, wenn gewisse Beziehungen zwischen den Koeffizienten bestehen. Dies ist sowohl eine Verallgemeinerung seines fruheren Gesetzes fur Bohrsohe Atom wide des Gesetzes von Pauli

Das Problem des Warmegleichgewichts in einem Strahlungsfeld in Anwesenheit von materiellen Gebilden kann aber nach den Methoden der statistischen Mechanik behandelt werden, unrtbhiingig von jeder besonderen Hypothese iiber den Mecha- nismus der Elementqrozesse, auf denen der Energieaustausch beruht. Wir gelangen so zu einer allgemeinen Beziehung, die giietig ist fur alle speziellen Hypothesen uber die Elementarprozesse und ihre Wahrscheinlichkeiten. Wenn es moglich ist, die thermodynamische Wtlhres~heinlichkeit fur irgend eine spezielle Verteilung der

(3) Debye, Phya. Z S . 24, 161, 1923. (4) Compton, Phya. Rev. 21, 483, 1923. (5) Pauli, Z S . f. Phys. 18, 272, 1923.

(6) Einstein und Ehrenfest, ZS. f. Phya. 19, 301, 1923.

106 S N Bose : The Man a d His Work

Quanten im Strahlungsfeld und die Wahrscheinlichkeit fiir irgend eine willkurliche Verteilung der Enerige unter den materiellen Teilchen niederzuschreiben, dann- ist die thermodynamische Wahrscheinlichkeit des groperen Systems, das Materie unc Strahlung urnfapt, einfach das Produkt der bieden Wahrscheinlichketien. Die Glei- chgewichtsbedingung lautet einfach, dab die resultierende Wahrscheinlichkeit ein Maximum aein mu@. Das Gleichgewichts-problem kann eomit in einer ganza ganz anderen Weise betrachtet werden. Statt geeignete Hypothesen uber Elementarpro- zesse, die das Plancksche Gesetz ergeben, aufzustellen, vermchen wir, die Relation abzuleiten, zu der alle Hypothesen fuhren mussen, wenn das Plancksche Gesetz fur das Strahlungsfeld und das Maxwellache Gesetz fur die materiellen Teilchen gelten

sollen. Die Beziehung, die wir suchen, ist offenbar aquivalent mit der der zuvor aus der Bedingung, dab die thermodynamischche Wahrscheinlichkeit fur das gro/3 ere System ein Maximum ist, aabgeleiteten. Um das Problem in der angedeuteten Weise zu lusen, mussen wir den Ausdruck fur die thermodynamische Wahrachein- lichkeit fur die Strahlung und fur die materiellen' Teilchen haben.

Therrnodynamische Wahrscheiqlichkeit fur das Strahlungsfeld. Sie wurde in einer fruherent Arbeit7) abgeleitet. Wenn die Zahl der Quanten im Spektralbereich v und v+dv gleich N d v ist, so ist sie :

87rva WO A y = -T V d v bedeutet. Es ist leicht zu sehen, dap dieses Wahrscheinlich-

keitsgesetz zum Planckschen Gesetz fuhrt, wenn wir die Bedingung aufstellen, da@ W bei gegebener Energie ein Mximum sein mu&

Thermodynamische Wahrscheinlichkeit fur materielle Teilchen. Man knnn sic unschwer finden. Wir wollen unsere Voraussetzungen ein wenig verallgemeinern, um sowohl den Fall Bohrsoher Atome mit diskreten Energienitreaus als auch den Fall der Verteilung der Translationsenergie bei materiellen Teilohens ) zu umfassen. Wir nehmen an, der Phssenraum sie in Elementargebiete geteilt. Zu jeder Elementarzelle gehort eine bestimmte Zahl g, die die Wahrscheinlichkeit angibt, du/3 irgend ein Teilohen sich darin befindet. Die g sind im algemeinen einander gleich, ausgenommen den Fall Bohrscher Atome. Die thermodynamische Wahrscheinlichkeit itgend einer willkurlichen Verteillung n,, nz use uber die versohiedenen Zellen ist :

(7) Der Verfaamr, emheint im Phil. Meg. Siche auch Debye, Ann. d. Phya. 83, 1427, 1910.

(8) Bohr, 28. j. Phg8. 18, 117, 1923.

Warmeglei'chgewicht i m Strahlungsfeld. . . 107

Die Bedingung des Wahrscheinlichkeitsmaxirnums bei gegebener Gesamtenergie uncl Zahl cler Teilchen ist :

E -- n, = Cg, e K T '

Jetzt sind wir in der Lage, die thermodynamische Wahrscheinlichkeit fiir das gesamte System zu berechnen. Sie ist offenbar :

und EnT = N .

Beim Gleichgewicht ist W ein Maximum, wobei die Gesamtenergie und die Zahl der Teilchen als gegeben anzusehen ist.

Wir betrachten den ElementarprozeP :

n, verwandelt sich in n,- 1 und n, in n,+ 1 ; NV, , Nvp . . . vermhdert sich urn 1

wid NV1,, Nv2, ... wachst um 1.

Die Bedingung der Statjonaritat von W verlangt, dap W sich dabei nicht lindert.

Der betrachtete Elementarprom kann aufgefapt werden als der ubergang eines Teilchens vom r-ten zum 8-ten Zustand. Die Veranderung, die da$ Strahlungsfeld dabei erleidet, kann angesehen werden als das Ergebnis einer Zerstreuung nach einem Zussmmensto/3.

Die gesuchte Bedingung lantet :

wo Xhv' -Chv+E,-E, = 0.

Diese Gleichung ist offenbar aquivalent mit

wo

Dies kann auch geschrieben werden :

n,nblpv,n(a; +b; e', ) = n , m p',,(a,+b,p, 1,

... (I)


wenn Sr = Sr 9

Das ist die verallegemeinerte Gleichung, die fur Elementqrozesse von der Art, wie sie Einstein und Ehrenfest9) betrachetet haben, gilt. Nach der hier gegebenen Ableitung ergibt sic& die Beziehung als eine Folgerung aus der Bedingung, dap beim Gleichgewicht die Wahrscheinlichkeit ein Maximum sein mu/?. Wenn aber der Mecha- nisms des Elementarprozesses uns gestattet, die Gleichung direkt niederzuchreiben

fuhrt die Substitution des Wertes fur dm Verhllltnis 3 zum Planckschen Gesetz. n8

Es is jedoch klar, daa Gleichung (I) fundamentaler ist als die transformierte Form von Einstein. Die nachfolgende Betrachtung wird zeigen, dap eine einfache Uberlegung den Wert des Wahrscheinlichkeitskoeffizienten in der gewiinschten E'orm liefert.

Der Fall des Bohrschen Atoms.

Das ist der Elementqroze/?, den Einstein'lo) in seiner fruheren Arbeit betrachtet hat.

Die fundamentale Gleichung (I) reduziert sich hier auf

Einsteins urspriingliche Gleichung lautet :

Nimmt innn ferlicr an, da/l

so folgt das Plancksche Gesetz.

Um seine Gleichung (I) abzuleiten, nimmt er an, dap Atome von htikeren Ener- gieniveaus zu Zustanden niederer Energie auf zwei Wegen iibergehen konllen :

1. Eine Art von spontanem Ubergang, wie bein einem radioactiven Prozep, dessen Wahrsoheinlickkeit uniibhangig vom Zustand des Strahlungsfeldes ist.

(9) Einstein und Ehrenfest, a. a. 0. (10) Einstein, Phye. 28. 18. 121, 1917.

Warmegleichgewicht im Strahlungsfeld. . . 109

2. Ein durch ,die charakteristische Strahlung induzierter ubergang, dessen Wahrscheinlichkeit vom Zustand des Feldes abhlngt .

Der U'bergang von einsm niederen zu einem hiiheren Energieniveau erfolgt durch Absorption, deren Wahrscheinlichkeit vom Felde abhiingt.

Weiterhin werden gewisse Benziehungen zwischen den Koeffizienten postuliert, damit das Plancksche Gesetz herauskommt. Diese Beziehungen konnen nicht auf einfache Weise begriindet werden.

Das gleiche Problem k a m auf die folgende Weise behandelt werden, bei der viele von den vorkommenden willkurlichen Annahmen beseitigt werden.

Wir setzen auch voraus, dap der dbergang von niederen zu hoheren Niveaus stets durch Absroption von Quanten einer charakteristischen Frequenz erfolgt.

A. Der ubergang von hijheren zu niederen Niveaus ist eine Art von spontarier h;nderung, dessen Wahrscheinlichkeit unabhangig vom Strahlungsfeld ist. Die zweite Hypo- these von Einstein (negative Einstrahlung) ist nicht erforderlich. B. Wir wollen

versuchen, den Wahrscheinlichkeitskoeffizienten fur ubergang A zu berechnen.

Die Zahl der Elementargebiete, die den in Betracht kommenden Quanten entsprechen, ist A,. Die Gesamtzahl der vorhandenen Quanten ist N d v . Diese sind aber nicht gleichfijrmhg uber den Phasenraum verteilt. Sei p, die Zahl der leeren ZeUen,p, die rnit 1 Quant, p1 die mit 2 Quanten usw. besetzten. Damit die Strahlung und daa materielle Teilohen in Wechselwirkung treten konnen, mup is dem Ellementargebiet sein, das von einem Qumt besetzt ist. Das besondere Elemeqtargebiet, in dem sich das betrachtete Molekul behdet , wird, wenn wir es lange genug beobaclhten, alle moglichen Zustiinde durchlaufen : d.h. manchmal ist es leer, manchmal wird 1 Quant vorhanden sein, manchmal 2 usw. Die Dauer dieser Ereignisse wird Schlei~lich proportional zu den Gleichgewichtswerten von p,, p,. . . Wenn r Quanten und ein Teilchen zugleich vorhanden sind, dann gibt es entweder einen Energieaustausch oder gar nichts. Somit sind r f l verschiedene Ereignisse mogboh, niimlich Energieaustausch von 1 Quant, 2 Quanten usw. oder kein Austausch

Daher ist die Zahl der moglichen Fiille :

Die Zahl der Falle, in denen Wechselwirkung eintritt, ist :

p1+2p,+ ... = N,dv, = Xrp,.

Somit ist die Wahrscheinlichkeit einer Wechselwirkung


Nun mussen wir die sgezielle Art der Wechselwirkung in Rucksicht ziehen, d.h. die Absorption, die zu unterscheiden ist von Zerstreuung. Die Wahrscheinlichkeit hierfiir sie 8.

Die Wahrscheinlichkeit der Absorption ist :

Der Wahrscheinlichkeitskoeffizient fur einen Ubergang vom Typus B ist offenbar

irgend eine Konstante, die den ubergang von dem speziellen hijheren Zustand zu dem speziellen niederen Zustand bestimmt. Dime Konstznte sei a.

Die Gleichgewichtsbedingung kann dann geschrieben werden :

was mit der fundamentalen Gleichung (I) ubereinstimmt, wenn wir annehmen, dap

srP = s,a.

Es ist klar, dab die Substitution von 3 zum Planckschen Gesetz fiihrt. n,

Der Fall von Pauli.

Er ist einfach zu behandeln. Die Wahrscheinliohkeit einer Wechselwirkung ist

wire vorhin . p;, sei der .Wahrscheinlichkeitsko8ffizient fur Zerstreuung m+ 1)pr van V, in v,. fads cine Wmhselwirkung statthat. Fur die inverse Traneformation

seien die entspreohenden Koeffizienfen r p und &. w + UP; Beim Gleichgewicht ist also :

Nehmen wir ferner an, dab Pi, = fi' so erhalten wir die fundamentale Glei- chung (I).

Der Fall von Einstein urid Ehrenfest.

Die Wahrscheinlichkeit einer gleichzeitigen Wechselwirkung ist di~s Produkt der Einzelwahrscheinlichkeiten, also gleich :


Sl S2 " ' Analog sei P , t ,I die Wnhrscheinlichkeit fiir den speziellen Akt der Zerstreuung,

1 9 . . .

und cin ihlichcr Ausdruck gelte fiir die inverse Streuung, woraus sich das ubrige lcicht ergibt.

Bemerkungen. Aus unserer Rechung gelit hervor, da /3 die Wahrscheinlichkeit einer Wechselwirkung eines Teilchens nlit einem Quantum in einem Strahlungsfeld

2\r, dv A.+N,,dv

nicht einfach proportional zur Zahl der vorhandenen Quanten ist, wie

man auf den ersten Blick vermuten kiinnte. Es ist leicht zu sehen, dap diese Annahme zum Wienschen Gesetz frihren wiircle, wie Pauli erkannt hat1". Pauli sah sich deshalb genotigt, eine andere Form fiir den Wahrscheinlichkeitskoeffizienten nzuunehmen. Die von Pauli angenommene und von Einstein und Ehrenfest verallgemeinerte Form erscheint aber als ganz willkurliche Hypothese, da man sich kein einfaches Bild von dem Zustandekommen eines solchen Ausdrucks machen kann. Die andere hier vorgeschlagene Form ist siemlich einfach und kann auf Grund elementarer ~ b e r l e - gungen gerechtfertigt werden. Es wird auch die Notwendigkeit vermieden, Fela- tionen zwischen den Koeffizienten selbst annehman zu miissen. Bei der Ableitung des Wahrscheinlichkeitskoeffizienten fur die Wechselwirkung (oder Kopplung, wie Bohr sagt) wurde angenommen, dap selbst bei einem Zuss.mmenstop der Fall, dap keine Wechselwirkung eintritt, gerade so wahrscheinlich ist, wie der Fa1 irgend einer spezielen Kopplung Diese Annahme ist ein fundamentaler Punkt in der hier gegebenen Ableitung. Nach der klassischen Theorie wiirde man erwarten, da/3 irgend eine Wechselwirkung eintritt, wenn ein Quant mit einem materiellen Teilchen zusam- mentrifft. Eshandelt sich also um eine principielle Abweichung von der klassischen Theoire. Diese Hypothese ist aber (wie mir scheint) ganz Lhnlich wie die Hypothese, die man gewohnlich iiber die Stabilitat der stationaien Zustande macht. Die klassische Theorie lie pe uns erwarten, dap keine stationaren Zustande moglich sind, und dab stets Wechselwirkung oder Kipplung (Bohr) zwischen ~ t h e r und erregtem Atom und damit verbundene Strahlung vorhanden ist. Auf der anderen Seite mussen wir, um die spektrale Emission zu erklaren, den Fall als moglich annehmen, dap keine

Wechselwirkung (Emission) eintritt. So kommen wir dauz, wegen der Persistenz der stationaren Zustande jedem beliebigen stationaren Zustand einen Wahrsohein- lickkeitskoeffizienten oder eine mittlere Lebensdaur zususchreiben.

Die oben benutzte Hypothese, dap aich bei einem Zuszmnlensto/3 keine Wech- selwirkung eintreten kann, ist ganz analog der Annahme der Stabilitat der stationtiren Zustande, die so grundlegend fur die Bohrsche Theorie der Linienemission ist, und kann auf die gleiche Ursache zuriickgefuhrt werden : Die den materiellen Teilchen innewohnende Wahrscheinlichkeit der Persistenz ihrer stationaren Zustande. In


diesem Zusammenhang ist es interessant, zu bemerken, dap wir beim analogen Fall des Zusammensto/3es eines Elektrons und eines Atoms experimentell Fiille nachweisen konnen, wo bei Zusammenstopen das Elektron durch das Atom hindurchgeht, ohm die intraatomaro Bewegung oder seine eigene zu verandern.

Die Beziehung g,p = g,a . Die Koeffizienten, oder wie Bohrla) sie nemt, Gwichte g werden eingefuhrt, wenn

Valenzelektron im Atom eine entartet bedingt periodische Bewegung ausfuhrt. Diese numerischen Koeffizienten geben nach Bohr einfach an, auf wieviel verschiedene Weisen man von einer benachbarten nichhntarteten Bewegung zu der speziellen entarteten Bewegung als Grenzfall gelangen kann. Nehmen wir diese Koeffizienten einfach als proportional zur Zahl der ubergangsmijglichkeiten von einem bestimmten zu dem betrachteten Zustand an, so ist die obige Beziehung fast selbstverstlindlich. Es gibt g, Wege, auf denen ein ubergang von irgend einem r-ten zu dern betrachteten a-ten Zustand durch Absorption mijglich ist. 'Analog gibt es g, ubergange zu niederen Zustanden infolge von Emission. Der Wahrscheinlichkeitskmffizient hat dmn den Faktor g, der Wahrscheinlichkeitskmffizient der Emission din Faktor g,. W e n wir ferner mnehmen, da/3 die Wahrschelichkeit einer Transformation durch Strahlung auf einem bestimmten Wege die gleiche ist wie der Wahrscheinlichke- itskoeffizient der Emission, wenn das Atom in bestimmter Weise zu niederen Zustiinden ubergeht, so ergibt sich sofort die fragliche &latior~13).

Ioh halte Boses Hypothese uber die Wahrscheinlichkeit der Strahlungselementar- vorgmge aus folgenden Grunden fur nioht zutreffend.

Fiir drts statistische Gleichgewicht zwischen einem Bohrschen Zustmde zu einem anderen gilt, wie Bose dazgelegthat, die Beziehung

Daraus folgt, dq? die Wahrscheinlichkeiten fur die Ubergange r-8 und 8+?

der linken bzw. rechten Seite dieser Gleichung proportianal sein miissen. Die &r-

gangswahrscheinlicbkeiten fiir. ein Molekiil miissen sich also (wenn wir der Einfaohheit A halber die statistichen Gewichfe beider ZustiLnde gleich 1 setzen) wie ---I-: 1

4 + N " verhalten. Mehr kann aus der Kenntnis des thermodynamiohen Gleiohgewichta

(12) Bohr, 2.8 f. Phyu., a.a.0. (13) Vgl. in dieaem Z&enhang P. Hertz, Repert. d . Phyu., etetietieche M w h d , 1 . Bd.,

Tell 2, 8. 649.


nicht entonommen werden. Nach der von mir auftestellten Hypothese sollen diese Wahrscheinlichkeiten N , (d.h. der Strahlungsdichte) bzw. A,+N, proportional sein,

N" nach der Hypothese Boses A,,+N, brw' I'

Nach der letzteren Hypothese kann die aiij3ere Strahlung wohl einen ubergang von Zustande Z, kleinerer Energie nach dem Zustande Z, grijperer Energie bewirken,

nicht aber umgekhrt einen ubergang von Z, nach Z,. Dies widrespricht aber dem mit Recht allgemein anerkannten Grundsatz, daj3 die klassische Theorie einen Grenzfall der Quantentheorie darstellen miisse. Nach letzterer kann narnlich ein Strahlungs- feld auf einen Resqnator sowohl positive als auch negative Energie iibertragen (je nach der Phase), und zwar beides gleich wahrscheinlich. Die Wahrscheinlichkeiten

beider ubergange miissen also von der Strahlungsdichte, d.h. von N , abhlingen, im Gegensatz zu Boses Hypothese. Inwiefern die Qunatentheorie die klassische zum Grenzfall hat, hat Planck in der letzten Auflage seines Buches iiber Strahlungs- theorie ausfiihrlich erotert.

Zweitens mu te nach Boses Hypothese ein kalter Korper ein von der Strahlunge- dichte abhangiges (mit ihr abnehmendes) Absorptionsvermogen besitzen. Die Korper sollten in Kaltem Zustande "nicht-Wiensche" Stra'hlung schwgcher absorbieren ah weniger intensive aus dem Giiltigkeitsbereich der Wienschen Strahlungsformel. Dies ware bei ultraroter Strahlung heiper Lichtquellen ge* schon entdeckt worden, wenn es sich so verhielte.

Reprinted from Z Phpik 27, pp. 384-393, 1924 (Springer - Verlag, Heidelberg).

Thermal Equilibrium in the Radiation Field in the Presence of Matter

S. N. Bose in Ramna (India) (Received 7 July, 1924)

The probability of an arbitrary state of a system consisting of radiation and molecules is calculated and from i t the conditions for statistical equilibrium are derived. Further, new expressions for the statistical probability of elementary processes, appropriate for the interaction between radiation and matter, are proposed.

Debye1 has shown that Planck's law can be derived using statistical mechanics. His derivation is, however, not completely independent of classical electrodynamics, because he uses the concept of normal modes of the ether and assumes that for calculating

8 7c v2 the energy the spectral range between v and v + dv can be replaced by - Vd v

c3 resonators whose energy can be only multiples of hv. One can however show that the derivation can be so modified that one does not have t o borrow anything from the

8 x: v2 classical theory. - Vdv can be interpreted as the number of elementary cells in

c3 the six dimensional phase space of the quanta. The further calculations remain essentially unchanged.

Einstein has used another method. He considers the interaction between material particles and the radiation field. If one knows the energy distribution among the particles, then one can derive the law of black-body radiation from the condition of stationarity of this distribution when there is an exchange of energy between matter and radiation. This exchange is moreover a result of certain elementary processes. With a suitable formulation of properties and the probability of these elementary processes one can derive Planck's law, if the energy distribution among the particles is assumed to be known. In his first paper2 the material particles are Bohr atoms which can have only a series of discrete stationary states. he energy exchange occurs because of emission and absorption processes and the simultaneous transitions of atoms from one state to another. He has shown that Planck's law is obtained when the emission and absorption processes have certain probabilities. Recently the problem of thermal equilibrium between radiation and and free electrons has acquired added importance through the studies of Debye3, Compton4 and others, which deal with the scattering of radiation off electrons. pauli5 has discussed this problem in an interesting paper. As an elementary process he considers the scattering of an electron by a quantum which moves in a definite direction and has a frequency in a given spectral range and is converted into another quantum with a different frequency and moving in a different

Thermal Equilibrium in the Radiation Field. . 115

direction. Simultaneciusly an electron which moves with a certain velocity in a given direction changes the magnitude and the direction of its velocity. The laws of conservation of energy and momentum hold for all these processes. Pauli has shown that if the probability of this elementary process is (Ap + Bpp') dv where p and p' are radiation densities for frequencies v and v', A and B are independent of p and p' but depend on the nature of collision, this probability law leads t o Planck's law, provided Maxwell's distribution is valid for the electron gas. Einstein and ~ h r e n f e s t ~ have further generalized Pauli's result. They consider a process in which a molecule or an electron absorbs quanta hv, , hv2 , etc. and simultaneously emits hv; , hv; ... It is assumed that the emitted and absorbed quanta have definite directions. The magnitude and the direction ofthe velocity of the molecule under consideration also change a t the same time. Energy and momentum are however conserved. The probability law of Pauli is generalized as follows,

d Wl = L ! b , g , n ( a ; + b ; ~ ; ) d t , d 'CV2 = - ( a l + b , Q 1 ) n(b; 8 ; ) d t ,

for the direct and the inverse processes. They show that equilibrium is obtained if certain relations between the coefficients exist. This is a generalization of Einstein's earlier result for Bohr atoms and also of Pauli's result.

The problem of thermodynamic equilibrium of radiation in the presence of material particles can, however, be studied using the methods of statistical mechanics, independently of any special assumption about the mechanism of the elementary processes on which the energy exchange depends. In this way we obtain a general relation which is valid for all special assumptions about the elementary processes and their probabilities. If it is possible t o write down the thermodynamic probability for any special distribution of quanta of radiation and the probability for any arbitrary energy distribution among the particles, then the thermodynamic probability for the bigger system containing matter and radiation is simply the product of the two probabilities. The condition of equilibrium is simply that the resulting probability must be a maximum. The problem of equilibrium can thus be looked a t in a completely different way. We try'to derive the relation t o which all assumptions must lead when Planck's law for radiation and Maxwell's law for particles hold instead of making assumptions about elementary processes adopted t o give Planck's law. The relation which we are looking'for is clearly equivalent t o the previous one which is derived from the condition that the thermodynamic probability for the bigger system is a maximum. In order t o solve the problem in the way indicated, we must have the expressions for the thermodynamic probability for radiation and the material particles.

Thermodynamic probability for radiation :

This has been derived earlier7. If N , dv is the number of quanta in the frequency range

v and v + dv, then the probability is,


8x:v2 where A , = - Vdv. It is easily seen that this probability law leads to Planck's law

c3 if we impose the condition that for a given energy W should be a maximum.

Thermodynamic probability for particles :

This can be easily found. We want t o generalize our assumptions a little so that the case of the Bohr atom with discrete energy levels as well as the case of the distribution of translational energy of particles can be included. Let the phase space be divided in to cells. For every cell there is a probability g that a particle occupies it. The g's are in general equal except for the case of Bohr's atoms. The thermodynamic probability for any arbitrary distribution n, , n, etc. among the different cell is,

N ! yy1g;z.. . n , ! n,!

The condition that the probability be maximum for a given total energy and number of particles is,

E -- n,. = k T C g 4 .

Now we are in a position to calculate the thermodynamic probability for the total system. I t is clearly

where

VV = 17 A 8 + N , ! 17- s ; ~ N ! .g A,! .Nb! nT! '

Z N s h v , + ~ ~ T E T = E

and x n T = N.

At equilibrium W is a maximum, the total energy and number of particles being given.

We consider the following elementary process : n, changes t o n, - 1 and n, t o n, + 1 ; Nvl , N,;, ... reduce by 1 and NvJl , Nvr2 increase by 1.

The condition of stationarity of W requires that W does not change on account of this process.

The elementary process under consideration can be considered as the transition of a particle from the rth t o the sth state. The change which the radiation field undergoes can be looked upon as the result of a scattering after a collision. The required condition reads

where

Thermal Equilibrium in the Radiation Field. . .

The equation (I) is evidently equivalent t o

g, = g,. This is the generalized equation which holds for elementary processes of the type considered by Einstein and ~ h r e n f e s t ~ . According to the derivation given here the relation follows from the condition that for equilibrium the probability must be a maximum. However, if the mechanism of the elementary process allows us to write

down the equation directly, the substitution of the value for the ratio 2 gives Planck's n s

law. I t is however clear that the equation (I) is more fundamental than the original one

of Einstein. The following discussion will show that simple considerations give the value of the probability constants in their desired form.

The case of Bohr's atoms : This is the elementary process which Einstein had considered in his earlier paper2. The fundamental equation (I) here reduces to

Einstein's original equation states

n,. b, ~ , , d t = n, (a; + b; gv)cZ.t or

If we assume further that a; h 713

9,. b, = g, b; and -;- = 8 n - bl cs '


then Planck's law follows : To derive his equation (I), Einstein assumes that atoms in higher energy levels make

transitions to lower energy levels in two ways : 1. A type of spontaneous transition as in a radioactive process, whose prob-

ability is independent of the state of the radiation field. 2. A transition induced by the characteristic radiation, whose probability de-

pends on the state of the radiation field. The transition from a lower t o a higher energy level is the result of an absorption

whose probability depends on the radiation field. Further, certain relations between the coefficients are postulated so that Planck's law follows. These relations cannot be justified in a simple way.

The same problem can be treated in the following way removing many of the arbitrary assumptions.

We also presume that the transition from a lower t o a higher state takes place through an absorption of a quantum of characteristic frequency.

A. The transition from higher t o lower levels is a spontaneous change, whose probability is independent of the field of radiation. The second assumption of Einstein (negative radiation) is not necessary.

B. We shall try t o calculate the probability coefficients for the transition A. The number of cells corresponding to the quanta under consideration is A,. The total

number of quanta present is N , dv. They are however not distributed evenly in phase space. Let p, be the number of empty calls, p , with 1 quantum, p , with two quanta etc. In order that the radiation and a particle may interact, it must be in a cell occupied by a quantum. The particular cell which the molecule under consideration occupies will pass through all possible states if we observe it long enough, that is, sometimes it is empty, sometimes occupied by 1 quantum sometimes by-2 quanta etc. The length of these events will be finally proportional to the equilibrium values of p, , p , . When r quanta and a particle are together, then either an exchange of energy takes place or nothing happens. Therefore r + 1 different events are possible, namely energy exchange of 1 quantum, 2 quanta etc. or no exchange. Therefore the number of possible cases is

The number of cases in which interactions occur is,

Consequently the probability of an interaction is,

Now we must take into consideration the special nature of the interaction, that is, absorption which is to be distinguished from scattering. Let the probability for this be p. The probability for absorption is

Thermal Equilibrium in the Radiation Field. . .

P z W - - N, d l 1 I@ + 1 G A, + ~ , d v .

The probability coefficient for a transition of type B is evidently any constant which determines the transition from the particular higher state to a particular lower state. Let this constant be a.

The equilibrium condition can then be written as

N,, d v = an,, *"A,+ Nvdv

which agrees with the fundamental equation (I) if we assume that

n r I t is clear that substituting the value of - ,>one gets Planck's law.

n s

Pauli's case. This can be easily discussed. The probability of a n interaction is a s before ' r p r . Let P :. be the probability coefficient for scattering from vs t o v,, if there C ( r + l ) p r is a n interaction. Let the corresponding coefficients for the inverse process ' r P r1

and p . Then a t equilibrium C ( r + l ) p , ,

Let us assume further that P z . = P z'; then we obtain the fundamental equation (I).

Einstein and Ehrenfest's case. The probability of a simultaneous interaction is the product of individual prob-

abilities and therefore equal to

Analogously let P :;, S,'2;.;,, be the probability for scattering and let a similar expres-

sion hold for the inverse scattering. The rest then easily follows. Remarks.

N , dv Our calculations show that the probability of an interaction between a

A, + N , dv

particle and a quantum in a radiation field is not simply proportional to the number


of the quanta present as one would presume at first sight. I t is easy t o see that this assumption leads t o Wien's law, as realized by pauli5. He found i t necessary t o assume another form for the probability coefficients. However the form assumed by Pauli and generalized by Einstein and Ehrenfest appears to be completely arbitrary because one cannot easily see how such an expression can be derived. The form suggested here is quite simple and can be justified on the basis of elementary considerations. The necessity of assuming relations between the coefficients themselves is also avoided. I t was assumed in deriving the probability coefficients for the interaction (or coupling, as Bohr says).that even in a collision no interaction is as probable as the occurence of any special interaction. This assumption is a fundamental point in the derivation given here. From the classical theory one would expect that there is some interaction whenever a quantum and a material particle come tqgether. Therefore here i t is a question of a departure from classical theory. This hypothesis is (as i t appears t o me), however, very similar to the assumption one usually makes about the stability of stationary states. From classical theory one would expect that no stationary states are possible and that an interaction or coupling (Bohr) between ether and the excited atom and the radiation connected with i t is always present. On the other hand, in order to explain spectral emission we must assume it t o be possible that no interaction (emission) occurs. Because of the persistence of stationary states we are therefore led to assign to every stationary state a probability coefficient or a mean lifetime.

The assumption made above, that even in a collision no interaction need occur, is quite analogous to the assumption of the stability of stationary states which is so fundamental to Bohr's theory of line emission and can be traced back to the same origin-the probability of existence of the stationary states of the particles. I t is interesting to remark in this connection that in the analogous case of a collision of an electron with an atom, experiments show that the electron goes through the atom without changing either the interatomic or its own motiod.

The relation g, P = g, a . The coefficients or as Bohr calls them, the weights g, are introduced when the

valence electron in the atom executes a finite degenerate periodic motion. These numerical coefficients give, according to Bohr, in how many ways from a neighbouring non-degenerate orbit one can arrive a t the special degenerate orbit as the limiting case. Let us assume that these coefficients are simply proportional to the number of possibilities of transition from a definite state t o the state under consideration ; then the above relation is almost obvious. There are g, possible ways for a transition from an rth state to the given sth stqte through absorption. Similarly therekre g, transitions to the lower states as a result of emission. The probability coefficient P has then the factor g,, the probability coefficient for emission the factor g,. If we further assume that the probability of a transformation through radiation in a definite way is the same as the probability coefficient for emission when the atom makes transitions to lower states in a definite way, then the relation in question follows immediatelyg.

Manindra Physical Laboratory, Dacca University, June 14, 1924.

Thermal Equilibrium in the Radiation Field. . . 121

I consider Bose's hypothesis about the probability of elementarj radiation precesses as not appropriate for the following reasons.

For the statistical equilibrium between two states of a Bohr atom the following relation holds, as given by Bose :

I t follows that the probabilities.for the transitions r+ s and s+ r respectively on the left-and the right hand side of the equation must be proportional to each other. The transition probabilities for a molecule must therefore (if for simplicity we put the

statistical weights of both the states equal t o 1) behave as Nv

: 1. Nothing more A, + Nv

can be learnt from the knowledge of the thermodynamic equilibrium. According to my assumptions these probabilities should be proportional t o N, (i.e. the radiation den-

sity) and A, + N, respectively, whereas according t o Bose's assumption, N"

and 1 A" + Nv

respectively. According t o the latter assumption the external radiatian can indeed cause a

transition from a state Z, of lower energy to a state 2, of higher energy, but not the reverse transition from Z, to 2, . This, however, contradicts the generally and rightly accepted fundamental principle that the classical theory should be a limiting case of the quantum theory. According t o the latter a radiation field can transfer to a resonator positive as well as negative energy (depending on the phase) and indeed with equal probability. The probabilities of both the transitions must depend on the density of radiation, that is, on N, as oppsed t o Bose's hypothesis. Planck h b discussed in detail in the latest edition of his book on the theory of radiation t o what extent the classical theory is the limiting case of the quantum theory.

Secondly, according t o Bose's hypethesis a cold body should have an absorbing capacity dependent on the density of radiation (decreasing with it). Bodies in cold state should absorb "non- Wien" radiation t o a weaker extent than less intense radiation as defined by the range of validity of Wien's radiation formula. If the behaviour had been such, then i t would certainly have already been discovered in infra-red radiation from hot light sources.

A. Einstein

References (rearranged sequentially)

1. Debye,Ann. d . Phys. 33, 1427, 1910 2. Einstein, Phys. ZS. 18, 121, 1917.

3. Debye, Phys. 2 s . 24, 161, 1923. 4. Compton, Phys. Rev. 21, 483, 1923.


5. Pauli, ZS. f i Phys. 18, 272, 1923. 6. Einstein and Ehrenfest, ZS. f. Phys. 19, 301, 1923. 7. The autho;, to appear in Phil. Mag. See also Debye, Ann. d. Phys. 33, 1427, (1910). 8. Bohr, ZS. f. Phys. 13, 117, 1923. 9. See in this connection P. Hertz, Repert. d . Phys., Statistiche Mechanik, Vol. 1, Part 2,

p. 549.

English translation of Z physik 27, pp. 384-393, 1924 (Springer - Verlag, Heidelberg).

Messungen der Zersetzungsspannung in nichtwasserigen Losungsmitteln,

von

Susil Chandra Biswas und Sn. Bose,

Aus dem Englischen i ibcr t ragc~

(Mit 6 Figuren im Text.)

(Eingegangen am 18. 11. 26.)

~ b e r die Zersetzungsspannungen von Saizen in nichtwasserigen Losungsmitteln liegen nur sehr wenige Untersuchungen vor. P a t t e;l und Mott 1) haben die Kurven der Zersetzungsspannung von Lithium- chlorid in Alkoholen, Aceton und Pyridin unter Beriicksichtigung der Vorgange an Anode und Kathode aufgenommen. Kiirzlich haben Nii l ler und Duschekz), ferner Muller3). Hijlzl und andere die Zer- setzungsspannung bei Losungen von Silbernitrat in Pyridin untersucht und die Zersetzungsspannung des Lithiurns in Losungen von Lithium- chlorid und Lithiurnnitrat in Pyridin. Diese Untersuchungen ergaben ffir Pyridin betrachtlich hohere Werta der Zersetzungsspannung als fiir wasserige Losungen. Doch waren sie ohne rotierenden Kommutator im Stromkreis angestellt und die elektromotorische Gegenkraft erscheint daher durch das dem grossen obergangsw4derstand der an der Elek- trodenoberflache haftenden Schicht entsprechende Potential vergrossert. Ganz kiirzlich hat N e w b u r y in einer Veroffentlichung iiber ,,Uber- spunnung und ~ b e r ~ a n ~ s w i d e r s t a n d e ' ~ ~ ) festgestellt, dass ,,die gesamte, dern Durchgang des Stromes von Elektrode zu Elektrolyt sich entgegen- stellende Hemmung aus zwei verschiedcnen Teilen besteht, wovon der eine reversibel ist (reine ~be r spannung j und der andere irreversibel (Ubergangswiderstande). Die ohne Verwendung eines Kommutators an- gestellten Messungen der Uberspannung pflegen rnit einem Fehler von 0.5 bis etwa 2 Volt behafiei zu seinu. P e a r c e und F rances ) , sowie 1) Journ. Phys. Chcm. 8, 153 (1904); 12. 49 (1908). 2) 3fonatsh. f. Chemie 43, 76 (1922). 4) I'roc. Hoy. Soc. 443.4,-486 ('1925). 3) hlonatsh. f. Chenue 43, 429 (1923). 5) Journ. Phys. Chem. 18. 729 (1914).

124 S N Bose : The Man and H i s Work

Mort i m e r urld P e a r c e 1) habtn bei Silbernitrat Eiuzelpotentiale und Cesamtspannung in reinen und gemischten Losungsmitteln untersucht, und m a r in Methylalkohol, ~ t h ~ l a l k o h o l , Pyridin und Wasser, sowie in. binaren Gemischen dieser Stoffe. Dabei fanden sie, dass die Eigen- schaften der birlaren Gemische von Alkoholen mit Pyridin der Mischungs-

Fig. 1.

regel folgen, wahrend binare Gemische vom NTasser mit Methylalkohol, iithylalkohol und Pyridin meist ein ausgesprochenes Maximum oder Minimum der hier in Betracht kominenden Eigenschaften aufweisea. Dies deutet auf eine gegenseitige Beeinflussung der beiden Losungs- mittel hin.

1,) .Tourn. Phys. Chcm. 21: 275 (1917).

Messungen der Zersetzungsspannung . . 125

Im folgenden sollen die Ergetmisse dargelegt werden, die bei der Untersuchung der Entladurqssparinung von Chlorwasserstoff und einigen Alkalihalogeniden in r,einern Methy lalkohol sow oh1 als auch in lllischungen

' 0,4 0.6 I

olt

Fig. 2.

Fig. 3.


von Methylalkohol und Wasser erhalten wurden. Der Methylalkohol war von Merck acetonfrei bezogen, wurde etwa 12 Stunden iiber Kalk

Fig. 4.

steh.en gelassen und d a m noch- mals destilliert.

Die verwendete Versuchs- anordnung erhellt aus Fig. 1. Die Elektroden waren aus dik- kem Platinblech und mit Platin- schwarz uberzogen. Durch den rotierenden Kommutator C konnte der Strom fast 3300mal in der Minute gewendet werden, so dass die Polarisationsspan- nung der Zelle gemessen werden konnte, nachdem der elek- trolysierende Strom bei jeder Wendung nur wahrend 0.009 Sekunden die Zelle durchflossen hatte. Die Starke des durch die Zelle gesandten Stroms wurde durch ein sorgfiiltig ge- eichtes Galvanometer mit Nebenschluss gemessen, fiir die haheren


Stromsttirken wurde ein Prazisions-Mil1iamperemete.r verwendet, Die Ergebnisse sind aus den Fig. 2, 3, 4 und 5 ersichtlich, worin

sie als Stromspannungskurven niedergelegt sind. Die Kurven fiir die ~erselzun~ss~annungen in wiisseriger Losung (bei entsprechender Kon- zentration) sind ebenhlls eigeuen Messungen entnom men und stehen in guter ~bereinstimmun~ mit friiheren Arbeiten anderer Autoren 1).

Tabelle 1. XJ. . - - -

1 norm. K J in CH3. OH ( 1 norm. KJ in H20

0,132 norm. KBr in CH3. OH

Entlatlungsspan- Slron~sldrko nung in Volt 1 in 10-5 A ~ n p .

Tabelle 2. KBr.

I I

Enllndungss~)an- nung in Volt

Entladungs- spannung

in Volt

SLrotnsliirke in 10-5 Amp.

Slro~nsl iirko in 10-5 Amp

Entladungs- spannunp

in Volt

3.066 norm. K&. in CH3. 0 E

Slro~l~sli irkc in 10-5 Amp.

- Ihtladungs-

spunnung in Voll

1) Park in, ,,Practical methods cif E~ac t1~ochen :~ i s t r~~~ . zug5uglich i s l , kiinn ich die Scitcnzahl nieht angcbcn.)

Stron~stli~ltc in l(rJ Amp

(Da mir dies Bnch nicht mehr

I

S N Bose : The M a n and His Work

Tabelle 3. LiCZ.

1 norm. LiCl in CH8. OH (0.1 norm. G O 1 'in OH3. 08 Entlndungs-

spannung in Volt

Stromslarke n 10-5 Amp.

Enlladungs- spannung

in Voll

SLromsttirke in 10-5 Amp.

1 norm. LiCl in Hp0 - Entladungs-

spannung in Volt

Stromstlrke in 10-5 Amp.

Tabelle 4. HCI. Z e r s e t z u n g s s p a n n u n g d e r S a l z s i i u r e i n 0.5 n o r m . L i i s u n g e n .

V = Entladungsspannung in Volt, CL = Stromstarke in 10-5 Amp.

Messungen der Zersetzungsspannung . .

Tabelle 4 (Fortsetzung).

Die -\'ei*suchsergebnisse. Kaliumjodid (vgl.' Tabelle 1 und Fig. 2).

Kaliumjodid wurde . in ungefahr normaler methyialkoholischer Losung verwendet. Die gef'undelie Zersetzungsspannung liegt bei nur 0.32 Volt. Die Anodenfliissigkeit wurde durch die hef'tige Jodabschei- dung schnell angekrbt. In wasseriger Losung wurde die Zersetzungs- spannung von normalem Kaliumjodid bei 1.12 Volt gefunden ( P a r k in fand 1.14 Volt).

Hsliumbro.mid (vgl. Tabelle 2 und Fig. 3). Etwa 0.1 und 0.05 uorm. Lijsungen von Kaliumbromid in reinem

Methylalkohol ergaben eine Zersetzurlgsspannung von 0.68 bzw. 0.71 Volt. ~ l~ens i ch t i i ch steigt hier die Zersetzungsspannung mit der Verdunnuug. i n wasseriger Losung liegt der Wert fiir 0.1 norm. I(a1iumbromid bei 1.254 VoIt (nach P a r k i n in noranuler Lijsung bei 1.61 Voltj.

I Lithiumchlorid (Vgl. Titbelle 3 und Fig. 4). Bei 0.1 norm. Lithiumchlorid zeigt die Kurve nur einen scharf'eu.

Knickpunkt und zwar bei etwa 1.2 Volt; bei einfach norm. Losung, ebenfalls in reinem Methylalkohol, tritt bei etwa 1-90 ein zweiter Knick auf. Nun ergibt eine einfach normale Losung von Lithiumchlorid in1 Wasser bei 1-95 Volt einen Knickpuukt. So kiinnte man vielleicht an-


nehmen, dass in der einfach norrnalen Lithiumchloridlosung in reinem Methylalkohol Spuren ron Wasser enthalten sind, die durch das stark hygroskopische ~ a l z selbst hinein gebracht wurden.

Chlorwasserstoff (vgl. Tabelle 4 und Fig. 5). Chlorwasserstoff wurde in etwa 0.6 norm. Li5sung untersucht und

zwar sowohl in reinem Methylalkohol und reinem Wasser, als auch in Mischungen der beiden von verschiedenem Prozentgehalt (10, 30: 50, 70 und 900/,, Wasser). In reinem Methylalkohol wurde die Zer- setzungsspannung bei 0.41 Volt ermittelt. Bei reinem und. igem Methylalkohol zeigte sich nach Erreichung der Zersetzungsspannung bei 0-4 bzw. 1-0 Volt deutlich die Wirkung des Chlors auf die Anode. Aus Fig. 5 ist die ~ n d e r u n g der Kurvenform bei den Gemischen deutlich zu ersehen.

Erijrterung des Einflusses von Gemischen. Beim Chlorwasserstoff hat der Zusatz von Wasser zu b1ethylalkohol

einen sehr deutlichen Einfluss auf die Zersetzungsspannung dieser Saure. Nun werden durch den Zusati' eines zweiten Losungsmittels die Eigen- schaften der Ionen des Elektrolyten immer dann sehr verandert, wenn zwischen ihnen und dem ersten Lijsungsmittel eine engere Bindung moglich war. Dies hat Krausl) an der Leitfahigkeit von Saurelosungen in Alkohol festgestellt, und zwar bei Zusatz von Wasser sowohl als von Salzen, die zur Komplexbildung neigen. Steigt der Prozentgehalt des Wassers in den methylalkoholischen Losungen der Sake bis zu 300/, und mehr, so tritt nach dem Knickpunkt bei etwa 1.0 Volt noch ein zweiter zwischen 1.3 und 1.33 Volt auf, der der Zersetzungsspau- nung der Salzsaure in reinem Wasser entspricht. Flir l0O,l0 igen Wassergehalt liegt die Zersetzungsspadnung bei 1.0 Volt und fiir reinen hIethylalkoho1 bei 0.41 Volt, wobei die Salzsaure in allen Fallen

1) ,Eigenschaften von Sjstcmen mit elel:trischer Leitf&Iiigkeit." S. 176 bis 184: Leitfahigkeit von Elektrolyten in Losun,nsn~ittelge1nisc1)~1i. Zeitaohr. f . physik. Chcmic. C,LX\-.


0.5 normal war. Demnach anclert sich die Zersetzungsspannung der Salzsaure nicht fortschreitend rnit steigendem Wassergehalt, sondern die Mehrzahl der Kurven zeigt zwei ausgepriigte Knickpunkte bei 1.0 und 1.33 Volt. Da in reinem &Iethylalkohol der h i c k bei etwa 0.41 Volt liegt, muss man den bei 1.0 Volt wohl der Bildung eines Komplexes in d e ~ n Losungsmittelgernisch zi~schreiben~

Der Einfluss des Wasserzusatzes auf Lithiumchlorid in Pyridin ist von P a t t e n und M o t t 1) untersucht worden. Ihre Resultate sind _nicht massgebend, da bei ihren Versuchen die Elektroden sich nlit einer isolie- renden Scllicht iiberzogen hatten. Die von ihnen erhaltenen Kurven entsprechen in ihrer Gestalt den hier wiedergegebenen. M iill e r , H o 1 z 12) und andere untersuchten die Entladung des Lithiums inlosungen von Lithium in Pyridin mit 5 und Wasser. ' Die erhaltenen Kurven waren vielfach geknickt und unregelmassig., Sie schreiben dies der Abscheidung von Lithiumhydroxyd zu. Lithiumchlorid geht mit Wasser iiberdies bekannterrnassen Molekularverbindungen ein, demnach not- wendigerweise auch mit einer Lijsung von Methylalkohol und Wasser3).

Der Einfluss des Wasserzusatzes auf methylaikoholische Losungen von Kaliumjodid und Kaliumbromid kann die Gestalt der Kurven nicht sehr andern, da -die Ionen dieser einfach gebauten Salze schwerlich mit dem stark assoziierten Methylalkohol zur Komplexbildung neigen werden. Die Leitfahigkeit wurde jedenfalls durch Wasserzusatz nicht stark veriindert.

Erorterung der Stroru~pannnngskurven. Die Stromspannungskurven in alkpholischer Losung zeigen samt-

lich einen einzigen charakteristischen Knickpunkt, der ziemlich scharf ausgepragt ist. In wasserigen Losungen wurden zwei deutlich verschiedene Knickyunkte festgestellt, wovon der eine, und zwar der bei der kleineren Spannung, nicht mit sichtbarer Zersetzung parallel geht, was bei dem zweiten aber untriiglich der Fall ist.


Die We, te der Entladungsspannung wurden in Xlkohol sehr viel niedriger gefunden als in Wasser. Andere Untersuchungen an Pyridin- lijsungen ergaben viel hohere Werte als in wasseriger Losung. AI iill e r gibt fur 1 norm. Lithiumchlorid in Pyridin 3.8 bis 4.0 Volt als Zer- setzungsspannung an und fur 0.1 norm. Losung 4.5 Volt, P a t t e n und No t t fanden 4.0 und 4.15 Volt als gesamte Polarisationsspannung der Zelle in Pyridin- bzw. Acetonlosung, wahrend der von uns in Methyl- alkohol gefundene Wert fur 0.1 norm. Lithiumchlorid nur' 1.2 Volt betragt.

Fiir die hier untersuchtec Substanzen zeigt sich in methvlalkoho- lischer Losung die gleiche Reihenfolge der Zersetzungsspannung wie in wasseriger, sie nehmen in der Reihe LiCl, KBr, RCI, KJstandig ab.

Znsrtmmenfassnng. 1. Fur Kaliumjodid, KaliumBromid, Lithiumchlorid und Chlorwasse~~

stoff wurden in reinem llethylalkohol Stromspannungskurven aufgenommen, die Zersetzungsspannungen bestimmt und die Kurven mit denen in wasseriger Losung verglichen

2. Bei Lithiumchlorid wurde festgstellt, dass schou die Gegenwari von Spuren Wassers die Gestalt der Zersetzungskurw gndert.

3. Es zeigte sich, dass die Werte fiir die Zersetzungsspannung von Losungen in reinem Methylalkohol tiefer li'egcn als in wasserigen Losungen, aber dieselbe Reihenfolge zeigen, namlich absteigend von Lithiumchlorid iiber Kaliumbromid zu Chlorwasserstoff und Kaliumjodid.

Reprinted from Z Phys Chem 126, pp. 442-451, 1927 (Johnson Reprint Corporation, NY).

Measurements of the Decomposition Voltage in Non-aqueous Solvents

Susil Chandra Biswas and S. N. Bose (The text contains 5 diagrams)

(Received on 18.11.26)

Very few findings are available on the decomposition voltage of salts in non-aqueous solvents. Patten and ~ o t t ' have recorded curves for the decomposition voltage of lithium chloride in various alcohols, acetone and pyridine, taking due consideration of the process taking place a t the anode and the cathode. In the recent past Mueller and ~ u s c h e k ~ and then ~ u l l e r ~ and Hoelzl and others examined the decomposition voltage in the solutions of silver nitrate in pyridine as well as the decomposition voltage of lithium in solutions of lithium chloride and lithium nitrate in pyridine. These results for pyridine showed considerably higher values for the decomposition voltage than those for the aqueous solutions; They were, however, taken without the installation of a rotating commutator in the circuit ; i t seems therefore that the opposing electromotive force was increased by the potential corresponding to the large contact resistance of the layer sticking t o the surface of the electrode. Very recently Newbury in a publication about the Overvoltage and the Contact ~esistances* has established that 'the total barrier opposing the flow of current from the electrode to the electrolyte consists of two different parts : out of these one (the pure overvoltage) is reversible and the other (the contact resistance) is irreversible. The readings of overvoltage without using a commutator are usually afflicted with an error of 0.5 t o 2 volts.' Pearce and I?rance5 as well as Mortimer and pearce6 investigated the individual potentials and the overall voltage for silver nitrate in pure and mixed solvents. These examinations were carried out in methyl alcohol, ethyl alcohol, pyridine and in water as well as in binary mixtures of these substances. These investigations revealed that whereas the binary mixtures of alcohols with pyridine followed the rules for the mixtures, the mixtures of water with

Fig. 1 : (see the German original)

Ruhrer = stirring rod Rollen = pulleys

Motor = motor Kathode = cathode

Anode = anode Gefallsdraht = step-down coil

Kommutator Schlussel =

commutator switch

Kompensations anordnung - -

compensator arrangement


Galvanometer mit Nebenschulss shunted galvanometer milliammeter for bzw. Milliamperemeter zur Messung des = measuring the electrolysing elektrolysierenden Stromes current

methyl alcohol, ethyl alcohol and pyridine mostly showed a pronounced maximum or a minimum in the properties considered here. This fact suggests mutual influence of the two solvents. The results as obtained from the examination of the discharge voltage of hydrogen chloride and a few alkali halides in pure methyl alcohol as -well as in mixtures of methyl alcohol and water are presented below.

Fig. 2 and Fig. 3 : (see the German original) (Stromstkirke in = Current intensity in Amp

Fig. 4 and Fig. 5 : (see the German original)

(Strmstkirke in = Current intensity in)1w5 Amp

11. Nl2 HC1 in 90% methyl alcohol, etc. --

Acetone-free methyl alcohol was obtained from Merck. It was allowed to stand for about 12 hours over lime and was then distilled once more.

The experimental set-up used is clear from Fig. 1. The electrodes were made of thick platinum plate and were coated with platinum black. The current could be reversed almost 3300 times per minute with the help of the rotating commutator C. This enabled the measurement of the polarization voltage of the cell when the electrolysing current after every reversal had passed through the cell only for a period of 0.009 seconds. The strength of the current sent through the cell was measured by a carefully calibrated shunted galvanometer. A precision milliammeter was used for measuring stronger currents.

The results are clear from the Figs. 2,3,4 and 5, where they are set down as current-voltage curves. The curves for the decomposition voltages in aqueous solutions (with respective concentrations) were similarly obtained from appropriate measurements and they tally well with earlier works of other authors.'.

Table 1. KI

(For numerical values, see the German original.)

Discharge potential in volts

Current strength in lo-' Amp


Current strength in Amp

Measurement of the Decomposition Voltage. . . 135

Table 2. KBr I I

Discharge

in volts in Amp in 10'~ Amp in 10" Amp

0.132 N KBr in CH3.0H


Table 3. LiCl I I

I I 1 I I

0.066 NKBr in CH3.0H 0.132 N KBr in H 2 0

1 N LiCl in CH3 .OH



Table 4. HCl

1 I I I I 0.1 N LiCl in CH3 .OH 1 N LiCl in H 2 0



The decomposition voltage of the hydrochloric acid in 0.5 N solutions 5 V = discharge potential in Volts, a = current strength in 10- Amp.

The Results of the Experiments


Potassium iodide (Cf. Table 1 and Fig. 2) A nearly 1N solution of potassium iodide in methyl alcohol was used. The reading for the decomposition voltage lies in the vicinity of only 0.32 Volt. The anode liquid became quickly tinged by the heavy iodine deposition. The decomposition voltage of 1N potassium iodide in an aqueous solution was found to be 1.12 Volt (Parkin found i t to be 1.14 Volt).

50% CH3 .OH 100% CH3 .OH

Potassium bromide (Cf. Table 2 and Fig. 3) Approximately 0.1 and 0.5 N solutions of potassium bromide in pure methyl alcohol yielded decomp'osition voltages of 0.68 and 0.71 Volt respectively. Apparently the

Current strength in lV5 Amp

90% CH3 .OH


70% CH3 .OH


136 $ N Bose : The Man and His Work

decomposition voltage increases here with dilution. In the aqueous solution i t was in the vicinity of 1.54 Volt for 0.1N potassium bromide (according t o Parkin i t is 1.61 Volt for a 1N solution).

Lithium chloride (Cf. Table 3 and Fig. 4) In the case of a 0.1N lithium chloride solution the curve shows a sharp bend in the neighbourhood of about 1.2 Volt. In the case of a 1N solution in pure methyl alcohol, a second bend similarly appears in the neighbourhood of 1.90 Volt. A 1N solution of lithium chloride in water shows a bend in the vicinity of 1.95 Volt. One could therefore assume that there are traces of water present (coming from the strongly hygroscopic salt) in a 1N lithium chloride solution in pure methyl alcohol. On the contrary, the majority of the curves show two marked bends in the vicinity of 1.0 Volt and 1.33 Volt. Since in the case of methyl alcohol the bend lies in the neighbourhood of 0.41 Volt, one must attribute the bend in the vicinity of 1.0 Volt t o the formation of a complex in the mixture of solvents.

The influence of adding water to lithium chloride.in pyridine was examined by Patten and ~ o t t ~ . Their results are not decisive as the electrodes became coated in their experiments with an insulating layer. The shapes of the graphs produced by them correspond with the ones reproduced here. Mueller, ~ o e l z l ~ O and others examined the discharge of lithium in solutions of lithium in pyridine with 5 and 10% water content. The graphs thus produced contained several bends and were irregular. They attribute this fact to the deposition of lithium hydroxide. Moreover, i t is well known that lithium chloride enters to form a molecular compound with water and therefore necessarily also with a solution of methyl alcohol aad water".

The addition of water to the solution of potassium iodide and potassium bromide in methyl alcohol cannot influence the curve in such a way as to'change its shape very much, because the ions of these simply built salts will be scarcely inclined to form complexes with the strongly associated methyl alcohol. In any case the conductivity was not much changed because of the addition of water.

Discussion of the Current Voltage Curves The current voltage curves in alcoholic solutions show a single sharply pronounced characteristic bend. In aqueous solutions two distinctly different bends were established of which the one with the lower voltage does not go along with a visible decomposition which is unmistakably the case with the other one.

The values for the discharge potential were found to be very much lower in alcohol than those in water. Other studies on pyridine solutions have yielded much higher values than the ones for an aqueous solution. ' ~ u e l l e r states 3.8 t o 4.0 Volt to be the decomposition voltage for a 1 N lithium chloride solution in pyridine. For a 0.1 N solution he states i t to be 4.5 Volt. Patten and Mott found 4.0 Volt and 4.15 Volt to be the total polarization potential of the cell in pyridine and acetone solutions respectively. In contrast, the value for 0.1 N lithium chloride in methyl alcohol was found by us to be only 1.2 Volt.

The decomposition voltage for subsfances studied here show the same sequence in

Measurement of the Decomposition Voltake . . . 137

a methyl alcohol solution as in an aqueous solution. They decrease steadily in the sequence of LiC1, KBr, HC1, KI.

Summary 1. Current voltage curves were obtained for potassium iodide, potassium bromide, lithium chloride and hydrogen chloride in pure methyl alcohol. The decomposition voltage was determined ; the curves were compared with those for aqueous so1utio.n~. 2. I t was established for lithium chloride that even the presence of traces of water changes the shape of the decomposition voltage curve. 3. It was seen that the values for the decomposition voltage in solutions of pure methyl alcohol are lower than those in aqueous solutions but show the same sequence, namely descending from lithium chloride to potassium bromide, hydrogen chloride and potassium iodide.

References (rearranged sequentially)

1) Journ. Phys. Chem. 8, 153 (1904) ; 12, (1908). 2) Monatsh. f. Chemie 43, 75 (1922). 3) Monatsh. f. Chemie 43,429 (1923). 4) Proc. Roy. S ~ C . 443 A, 486 (1925). 5) Journ. Phys. Chem. 18,729 (1914). 6) Journ. Phys. Chem. 21,275 (1917). 7) Parkin, Practical Methods of Electrochemistry (Since the book is no more accessible

to me, I cannot give the page number). 8) Eigenschaften von Systemen mit elektrischer Leitfahigkeit (Properties of Systems with

Electrical Conductivity.), pp. 176-184 : Conductivity of Electrolytes in Mixtures of Solvents.

9) loc. cit. 10)loc. cit. 11)Kraus, loc. cit.

English translation of ZPhys Chem 125, pp. 442-451,1927 (Johnson Reprint Corporation, NY).

Beryllium Spectrum in the Region h 3367-1964. By S.N. BOSE, Professor of Physics, Dacca University, and S . K . MUKHERJEE, Assistant Lecturer in Physics, Dacca University*.

[Plates 11. & 111.1

T HE beryllium spectrum is remarkable owing to the small number of lines that have been observed in the visible and the ultra-violet. The earlier measurements are somewhat conflicting. Exner and Haschek

could not find some of the lines observed by Rowland and Tatnall t. Glaser $ investigated the spectrum by sparking between the metal tips, but he.also could not corroborate the earlier observations.

Recently Millikan and Bowen 5 subjected the spectrum to a thorough analysis by their method of hot spark, and extended it up to 1943 A.u. They classified some of the important lines, and gave the term values both of Be I1 and Be I.

It was with a view to study the spectrum thoroughly, under different conditions of excitation, that the present work was undertaken.

The instrument that we have used almost exclusively for the work is Hilger's Quartz Spectrograph, type E 1. Ln the near ultra-violet between the regions 2400-1850 it is almost an ideal instrument for spectrum analysis, combining a high dispersion with a high light-value. We have extended our observations to the visible region; but here, because of the low dispersion, the measurements were not more accurate than by '1 A.u., though we could very easily identify the lines and thus verify the observations of the previous workers.

Most of the observations were made on the arc spectrum, and obtained by feeding the nitrate or the metal in a carbon arc. We have used both ordinary and Hilger's extra pure carbon rods, and the presence of traces of iron has been an advantage in furnishing suitable standard lines in the different regions. In the extreme ultra-violet we have used copper arc and spark as comparison spectra, and in some cases the silicon lines almost always present in. carbon spectra were helpful in making accurate measurements.

The plates used were IIford Empress and Ordinary up to 2175. In the extreme ultra-violet we tried sensitization with machine oil, but the lines were a little broad. As our aim was to resolve very narrow doublets, we could only get the best results by using Schumann plates as supplied by Hilger.

The materials used were, at first, certain old samples of beryllium nitrate and metal from Merck's, which were found to contain lead, aluminium, and silver as impurities. Later, owing to the kindness of Professor Mark, of Badische Anilin Fabrik, we were enabled to get a different sample of the metal. But this was found to contain traces of rare earths, notably scandium and yttrium. By a comparison of the different samples the lines due to contamination were eliminated, and we give our results tabulated below (vide Table I.).

As may be seen, we could go almost to the same limit as Millikan and Bowen by their hot-spark method, and, incidentally, we have discovered that the following lines, 2351, 2175,2126, 2056, and 2033, given as singlets by Millikan and Bowen, are clearly doublets of approximate wave-number difference 2.6. Most probably these are triplets due to the triplet P-terms, and our spectrograph could only separate PI from P2 + P3. The wave-number difference thus measured is the distance of PI from the centre of gravity of P2- and P3- lines, and as such has a slightly higher value.The lines 1998 and 1964 just appearedps doublets, but owing to their hazy character the measurements of the doublet separation were not possib16: We could verify the earlier measurements of Rowland and Tatnall as well as the line of Glaser, viz. h 4672'9. Incidentally, we have discovered a few new lines whose wave-lengths are given in Table I. The line 3019 appeared as a doublet, the line 2986 as a triplet, and the lines 31 10 and 2738 as singlets.

* Communicated by the Authors. t Kayser, Handbuch d . Spectro. vol. v. $ Glaser, Ann. d . Phys. (4) Ixvii. pp. 73-88 (1922). 3 Millikan and Bowen, Phys. Rev., Aug. 1926.

Beryllium Spectrum in the Region W367-1964 139

We have made some observations by feeding a fair amount of metal or the salt in the arc, and we could observe that the lines 332 1,265 1,2494, and 2 175 with their components were reversed in some cases, thus showing that thep-level is probably the common origin of the lines as classified by Millikan and Bowen. In rare occasions, howe-ver, the line 3 13 1, usually attributed to Be 11, could be reversed. The line 2,348 (1 S-2P) is always reversed in the arc.

We have tried the spark spectra in air, where, owing to rapid oxidation, we could make very little progress. We have also tried sparking between the tips of the metal, enclosed in apartially evacuated vessel with quartz window; and even then we could not go beyond 2175. In this way,however, we have got an interesting band spectrum, which we believe to be due to beryllium and which consists of bands degraded to the red. The fine structures of the band have not yet been measured, and we give in Table 11. the measurements of the edges.

There does not seem to,be any line of Be between 4700 up to the extreme red. We are now continuing the work in the infra-red, and hope to be able to corroborate the solitary work of Theo Volk * in the region.

The following table gives the wave-lengths (in I.a, vac.) of the lines between h3367-1964, with the frequency difference between the components, and also the lines (in 1.a) of Rowland and Tatnalll, together with the few lines marked * observed by us as new :-

TABLE I.

Bose & Rowland & Bowen & Mukherjee Tatnall Millikan 6v

I. a. I. a, vac. 3367.55 3367.579 - -

*3110.91 - *3019.54 - *3019.33 - -

2986.63 - - I - 2986.44 2986.426 - 2986.08 2986.057 - 2898.26 2898.24Q

*2738.08 - -

The lines marked (a) appear in plates, but their measurements are unsatisfactory owing to the absence of standards in this region.

*Dissert. Tiibingen, 1924 (Dresden bei Teubner). 17 pp.


Bom & MUKHEEJEE. Phil. Mag. Ser. 7, VoL 7, Pl. II.

A. Carbon arc fed with Be Salt. B. Iron arc.

Beryllium Spectrum in the Region k3367-1964

BOBS & MUXHEWEE. Phil. Mag. Ser. 7, Vol. 7. P1. 111.


In the accompanying plates (Pls.11. and 111.) I. and 111. are contact-prints from Schumann plates; 11. and IV. (a) are microphotographic enlargements for showing the fine structure; rv . , contact-print from ordinary plate; V. and VI. are slightly enlarged prints to show the new lines.

The following table gives the wave-lengths (in La) of the edges of the band; those marked (a) are fully developed with their fine structures:-

TABLE 11.

1 .......... 2474.2 2(a) ...... 2446.0 3 (a) ..... 2419.2 4 (a) ...... 2325.1 5(a) ..... 2299.4

Reprinted from Phil Mag Ser 7 ,7 , pp. 197-200,1929 (Taylor and Francis, London).

Section of Mathematics and Physlcs.

President :-PROF. S. N. BOSE, M.Sc

Presidential Address.

TENDENCIES I N THE MODERN THEORETICAL PHYSICS.

The ultimate aim of Scientific inquiry is to arrive a t a minimum number of hypotheses which will explain the maximum number of facts. The hypotheses should obviously not contradict one another. At the present moment, however, we see two contradictory theories, in the domain of Physics. On the one hand we have the classical theory based on the dyna- mica1 laws of Newton. On the other hand we have the Quan- tum theory first introduced by Planck, which has been differently formulated by different Scientists ; while the Classical Theory explain satisfactorily all problems relating to motion and interaction of big masses as well as the problem of propagation of radiation, the Quantum Theory has succeeded with the help of a few principles in co-ordincbting a large amount of esperimental material aocumulated in the various domains of spectroscopy, X-rays, etc. It has succeeded best in a11 problems dealing with the ultimate constitution of matter, or in problems dealing with the interaction of matter with radiant energy. As a result of work of the last twenty-five years, we seem to be much nearer to understanding the prohlem of matter. The periodic classificcb- tion of Mendeljeff does not now appear. as an unexplained riddle and the huge amount of spectroscopic material can now be classified and explained with the help of a few fairly simple principles. The task that faces the Physicists to-day is how best to harmonise the seeming discord of the two theories, which are a t the present moment utilised to explain the physical phenomena. I t is a difficult task, and the first step towards fulfilment will be to place in a clear light the differences that characterise them: for this purpose i t is necessary t,o trace the history of the development of our physical ideas indicating the various points where fresh hypotheses had to be introduced before the next move towards progress could be made.

Theoretical Physics may be said to have begun its career as a Science with the formulation of the famous laws of motion by Newton. The mathematical formulation of this principle leads to a series of differential equations in which we equate certain quantities which depend upon the state and nature of the body under investigation with certain other quantities which we interpret as forces arising out of the


interaction of other bodies. The solution of these differential equations introduces a certain number of arbitrary constants which depend upon the initial state of the system investigated. These constants once known determine completely the subsequent hist,ory of the system under observation. Newton's original equations involved the use of the cartesian co-ordinate system, but very soon these equations were transformed into a form in which the arbitrary character of the co-ordinates was removed, and finally Hamilton and Jacobi introduced the characteristic function, which reduced the problem of solving the equations of dynamics to the solutioa of a single partial differential equation. Simultaneously, with the help of the concept of the Variational Calculu~, we arrived a t the celebrated Hamilton's Principle, by which the whole problem of dynamics was reduced to the problem of making a certain .integra.l an extretnum, within certain suitable limits.

The writing down of the set of differential equations, or of the single.partia1 differential equation of Jacobi requires however the knowledge of the laws of interaction of various bodies on one another. The exact formulation of these laws seem therefore to be the immediate aim of the earlier classioal Physicists. These laws once known allow us to apply to any problem the general dynamical methods, whereby the problems of Physics are reduced to problems of pure mathematics. In the laws of gravitation: in the law of Coloiimb and of Ampere, we have some classical examples ; in every case it is endeavoured to express the force in terms of the relative co-ordinates, and the velocities of the interacting masses.

The laws of dynamics were originally formulated to explain the motion of observable bodies. When with the progress of knowledge the discrete nature of the constitution of matter was evident, the natural endeavour has then been to extend to those ultimate particles (the atoms, protons, electrons), the same dynamical laws which have been so successfully applied to the study of big masses. As the aim of Physics is to explain the observed physical phenomena, in terms of the motion of the ultimate particles, a convietent appli~at~ion of the dynamical methods has been responsible for the whole structure of classical Physics.

Before the methods of classical dynamics could be applied important developments in two directions had to he made. The problem of specifying the law of interaction of the different particles, when we have to take account of the immense number of the constituent particle^ raised difficulties which were resolved and the problems made more amenable to analysis by the introduction of the conception of the Field. It was discovered that instead of basing the description of interaction on the various laws, (tormulated on the action-at-a-distance basis), a much simpler and a better treatment could be obtained by the

Tendencies in Modern Theoretical Physics

illtroduction of certain auxiliary magnitudes-The scalar or the vector potential, or The Electric or Magnetic intensities. These quantities vary with the position in space, as well with time. The characteristic functions determining the field were found ili the ca.se of electro-magnetism to satisfy a certain linear set of partial differential equations, whereas the re1 a t ' ion between the magnitudes introduced, and the reacting masses may be expressed by a certain type of equation of the second order. These auxiliary magnitudes thus came to be looked upon as having their seat in a hypothetical medium which was postulated to pervade all space, and the partial differential equations which connect then1 came to be looked upon as related in some way to the physical properties of the hypothetical medium, the Ether. By the inkoduction of the concept of energy and momentum density, the abstract conception of ether seemed to gain in substantial reality. This was further strengthened when i t was shown tha,t i t was possible to bring the partial differential equations in line with the usual dynamical equations, in as much as they appeared to be deducible from the Hamilton's principle if certain quantities were identified with the potential and the kinetic energies of the medium. Thus we come to the classical conception of Ether, and the electromagnetic equations of Maxwell which served as the starting point of the modern Electron-Physics. Gravitation, however, did not immediately fall in line with the other field theory, until the formulation of the generalised Relativity-Principle by Einstein. Of this we shall have occasion to speak, later on.

There was however another difficulty which stood in the way of immediate application of the classical methods to the problems of Phpics. The dynamical laws seem a t first applicable to the motion of the ultimate particles, which themselves however always escape direct. observation. What we observe is to be looked upon as the resultant of a large number of elementary events; and the measurable microscopically sensible magnitudes cannot also be regarded as determining uniquely the component microscopic elements. Thus, from the classical standpoint, the necessary magnitudes for uniquely characterising the state of the system remain largely undetermined, and the Physical laws connecting observable things cannot be regarded as immediately deducible from the dynamical la,ws, at any rate without the introduction of further hypotheses. The method of statistical mechanics was developed to tackle this fundamental difficulty. The earlier method consisted in regarding the ultimate particles as practically independent systems. Each system is characterised by cert,ain values of co-ordinates and momenta. Under their mutual interaction, Space and momenta mrdinates of the individual systems are supposed to vary within certain limits. The actual observable magnitudes are determined by the distribution of the systems among the


different physically possible states. Mathematically this distribution is visualised by the distribution of the represent.ative points of the individual systems in a phase space, which is supposed to be decomposed into a certain number of elementary phase-cells. The actual position of any particular representative point inside the phase-cell, is supposed to have no effect on the value of the phenomenologically obsel-vable magnitudes. The equilibrium-state of a physical system can thus be related to a particular distribution of the representative points in the phase cells which occurs most frequently, or remaiu longest during the period of observation. By this way the important physical idea of entropy can be connected wtih the probability of distribution in the Phase-Space, and important Thermodyna- mical laws can be looked upon es certain stat;istical laws which are necessary consequence8 of the very large number of individual components. When the resultu of classical dynamics are combined with this concept of statistics, we get certain very general laws, regarding the partition of energy. Though at first, these results seemed to a.greo with the facfs observed, subeequent and more exact experiments have shown them to be erroneous.

I have already referred to the equations of classical mechanics which determine the motion of a materid particle, as well as the field equations, which connect the vector magnitudes serving to describe the electro-dynamical field. Though it was attempted to bring these equations in line with the dynamical laws by showing that they also can be deduced from Hamilton's Principle, a fundamental difference between the two sets of equations was clearly brought out during the early part of this century.. The dynamical equations of Newton poaeess an invariant group of transformation, the Galilean Group, which expresses the equivalence of all inertial systems as frames of reference for the desqription of motion. The field equations of Maxwell however, have a different invariant group, the Lorentzian group. The space and time co-ordinates enter symmetrically into these equations, or rather the space-time symmetry can be brought out by introducing after Minkow~ki an imaginary time ict., as the fourth co-ordinate. The Lorent- zian group of transformation can be represented by a rotation of axes in a four dimensional orthogonal space, which leaves the invariant distance unaltered. By extending the conception of vector quantities of the usual three dimensional space to four dimensions, the field equations can be giren an invariant form. The recognition of this formal equivalence of the time and space co-ordinates u~hers in the ideas of relativity. This equivalence now-a-days is regarded as axiomatic, and furnishes a test which all physical laws have to satisfy in order to be exact. By generalising this conception further, and by giving up the condition of orthogonality Einstein was able to present


a field theory of gravitation. The two different field theories have however remained essentially distinct though attempts have repea,tedly been made to fuse them into one single unitary theory.

I have now indicated to you the principal steps by which the classical theory arrived from its first elementary stage to the present developed state. I shall now tell you of the difficulties t,hat arose successively and which led to the formulation of the quantum theory. The conceptions of energy and momentum have been early deduced from the dynamical laws and have played very important roles in the subsequent development of the Science. For example the Hamilton-Jacobi differential equation requires for the mathematical expression the writing down of total energy of the system in terms of t,he momenta, whereas the integrand in the Hamilton's Principle is the difference between the potential and the kinetic energy of the system. The introduction of the concepts of energy seems necessary for the applicability of the general dynamical laws, and we have seen how the field equations can be made compatible with the dynarnical laws by interpreting a certain magnitude as the energy-density of Ether. As a necessary coro1lar.v to the equations however i t follows .that 'there will be a continuous interchange of energy between the ultimate particles composing matter and the surrounding radiation field. The principle of equipartition of energy which follows as a necessary consequence of the. idea, leads us however to entirely wrong results when applied to the problem of equilibrium of radiation-field with matter. In order to explain the distribution of energy-in the black-body radiation spectrum, as well as to explain the problem of generation of radiation we have been compelled to make certain assumptions regarding the constitution of the radiation- field, and about the equilibrium states of material particles. which directly contradict our classical ideas. This has led to the introduction of the Quantum theorv in Physics. T'he energy in the monochromatic radiation field, has to be assumed to exist in definite quanta. The study of the constitution of matter has led us to the conception of the atom as a comparatively stable structure built out, of the nucleus and electrons. These electrons according to the earlier formulation of the quantain theory are to be supposed as rotating in certain characteristic orbits ; whereas such a constitution on t,he claseical theory will necessarily continually radiate and lose energy, we have to assume here that such a structure keeps generally intact its energy-content and also that there are a series of discrete radiation-free states characterised by a discrete series of values of the energy content. An interchange of energy can only t,akc place, when the atomic system passes from one dlst.inct state to another resulting in an emission or an absorption of radiation. The passage of t,he atom from a higher energy state t,o a lower is


associated with an emission of a monochromatic radiation whose frequency is given by the relation

where W, and W' are the characteristic energies of the two states.

Modern development of spectroscopy seems to indicate that the state of a system may be defined by the introduction of certain sets of quanta members, each set being associated with a definite constituent electron of the atomic system. No two electrons of the atomic system may have the same group of the quanta-numbers, and the successive building up of the different atoms may be looked upon rts a gradual increase in the number of electrons in the outer orbit, each electron coming to occupy a place characterised by n different group of quanta numbers.

These few principles which are so contradictory to the ideas of classical dynamics have proved to.be invaluable in the development of modern Physics. To them we owe the scientific claesification of the spectroscopic terms, explanation of the Periodic classification, etc., etc. The direct and simple way in which the different problems, which have long baffled the attempt of classical physics, find their explanations in the new theory shows that it contains a Large element of truth, and the most important problem of the present time is to find a way of re- conciling the two theories. It has became evident also that no one theory will he able to entirely replace the other. For example the conception of the field, which has been developed originally from the classical standpoint, seems to contain also a great element of truth. The large mass of experimental material connected with the problem of propagation of radiation, seems to find in the field theory a very simple and direct explanation, which appear so difficult to explain from the Quantum- theoretical standpoint.

The solution of the dilemma seems to lie in suitably uniting the field and the quanta-theories, which will ultimately form a more general theory of which the two theories may then be regarded as limiting cases. We have in recent years already some indications of tentative attempts in the same direction. One may attempt to visualise the radiation-field as simply determining the interaction of the different part4icles and give up the classical concept of energy and momentum-density in Ether. Consistent with our ideas of the quantum theory, we may regard that the energy in the field is only discretely distributed througbout the space, (somewhat after the ideas of J. J. Thomson). The motion of these quantas of energy may be supposed to be guided as i t were by the field, and it may be supposed to take place along the direction of the Poynting


Vector. The same field which guides the motion of the quanta may be also supposed to control the motion of the electrons in the different orbits. The nature of the field itself should be determined by some equations depending upon the distribution of the charges. Such a field theory will have as its task the explanation not only of the fundamental relation E=hv , but also of the occurrence of the discrete quanta-states and the different quanta-members.

That the idea of discrete energy-states is not inconsistent with a continuous field-theory seems t,o follow from the recent developments of the wave-mechanics (Schrodinger, Dirac). I n the classical theory the impossibility of the existence of stationary electron-orbits follows from the result that an accelerated electron gives rise according to it to an electromagnetic field such tha t the surface integral of the Poynting Vector a t a great distance does not vanish. The result is interpreted as indicating a continuous loss of energy from the moving electron which makes stat,ionary orbits impossible. One can think however, tha t if there are other electrons present the resultant field may be such that the electron may in the average gain as much energy from the surrounding field as i t loses by radiation, or in other words the resultant electro-magnetic field due to all the moving electrons may be such that, the integral of the Poynting-Vector vanishes. Such a hypothetical solution of the field-equations which make the field vanish a t infinity may be thus likened to a stationary vibration rather than to a divergent wave-train. I t is clear, however, that the distrit~utiou of the charges will have to be suitably made and their motio~is definitely related in phase with one another in order that such a thing may be possible. The electro-magnetic field may then be supposed t,o be in resonance after de Broglie. What Schrodinger succeeds in establishing may be likened to finding exactly such a law of distribution of charges. The quanta conditions then may be fittingly compared with the resonance conditions. Dirac seems to have succeeded in deducing a set of four linear partial differential equations which correspond t'o facts more closely. The four functions or rather eight of t,hem give us the distribution of the hypothetical charges and currents in the electro-magnetic field necessary to est,abliuh resonance- c~nditions. The field itself in these two theories determines also the distribution or rather the statistically equivalent distribution (in space as well as in velocity) of the charges in the field.

One may hope that the final solution of problem may be- found and the proper set of equations which will determine the radiation-less field may be determined with the help of the ideas of the generalised relativity theory. If an unitary field theory which could explain gravitation and electro-magnetism could be found, one would hope to obtain $hereby not only the solution of the quanta-problem, but perhaps also of a more fundamental problem, the relation of charge with gravitational mass, and explanation of the difference in mass of the proton and the electron.

Reprinted from Proc Ind Sci Cong 16, pp. 55-62, 1929 (Asiatic Society of Bengal).

ON THE COMPLETE MOMENT-COEFFICIENTS OF THE D~-STATISTIC.

BY SATYENDRA NATH BOSE PROFESSOR OF PHYSICS, UNIVERSITY OF DACCA.

The D2-statistic ww defined by Mahalanobisl to measure the divergence between two statistical populations, and the moment-coefficients were also calculated by him by approximative methods. Rajchandra Bose2 has found the exact distribution of the D2-statistie* which can be expressed in terms of Bessel functions. He has used the actual distribution function to obtain the moment-coefficients, and has shown that the results previously obtained by Mahdanobis are exact. In the present no& a recurrence-formula for the D2-statistic has been obtained directly without assuming the distribution, and certain properties of these moment-coefficients have been investigated.

2. Let (a,, a,, .... a,) and (a;, ad, .... a;) be the observed mean values in two samples Z, and Z, of size n, and nz reepectively drawn from two normal p;yariate correlated populations with mean values (a,, a,, . . . a,) and (a;, a:, . . . 4;) res$ctively. The distribution of the differences in observed mean values (a,-a;, a,-a;, . . . a,-a;) can be written in the form :-

where /I and /Iif (i, j = 1, 2, . . . p ) are certain functions of the population variances and co-variances whose values have been given by Rajchandra Bose3 and

Then the D2-statistic is defined by

* In this paper p and n have been used in the place of P and la in the earlier papers of P. C. Maha- lanobia and Raj Chandra Bose. The DT-statistic was intended to be and was defined as a quantity determined entirely by the sample values of the variates. Raj Chandra Bose hes investigated the exect distribution and 5. N. Bose the moment-coefficients of a modified form of the Dz-statistic in which the population values of the variances and co-variances have been substituted for the corresponding sample estimates.-Edilot, Sankhyii.

and

Complete Moment-coefficients o f the LI2-statistic

The population value of D2 is A2 which isdefined by

3. If we put

(a,-a!) = zi, and (ai-a:) = ti a 1

we can write

If we now use a linear transformation which changes the quadratic form

then the same transformation will change

where ( q l , qZ, . .. 77p) are connected with ( E l , k, . . . C p ) by the same linear transformation which connects (y , , y2, ... y,) and (x,, x2, ... xp)

i . j=p Also + S (&zitj) will be transformed to ( ylql+ y2712+. . . yprp) ... (3.5)

i j=1

We can then write 2 4p.D:: = y;+y;+ .. .Y,

Equation (2.1) can then be written in the form :-

-? [ D ; + A Z - Z D ~ . A COSOI

Const. x e xdV

where dV = dy,, dy2 ... d y ,


4. Let be the k-th raw moment of @, and let

where the eingle sign of integration etanb for the pfold integration.

Then = 0 4 ... (4.2)

we sleo write np/4 = t . . . (4.3)

Q - ~ ( ~ + A ~ - ~ D * . A coa 8)

and Mk = 1 9 . e .d V ... (4.11) 0

Then - ( t ~ f + A ~ - Z D ~ A COB 0.1 '3 = - 2t j ~ k ( ~ - o l ~ ~ e)a

aA d V ... (4.4)

-~(D~+A'-~DIA coe 0) = -2t.A.Mk+2t j' *COB 8 . e d V ... (4.41)

Therefore

A axk ax, Mk+l = AS. Aft+-. - -- t aA at

... (4.6)

6. Now mtnally Q

Therefore

Complete Moment-coefficients of the LP-statistic

Ale0 &(Df) = c M, = 1 and therefore

We can use the above initial values to calculate MI, M2, ... Mk etc., and hence with the help of equation (4.2) obtain the raw moments (about any arbitrary origin) of Ds which can be written in the form :-

6. It will be convenient to use new variables

... ... u = A2, and s =. l/t (6.1)

Equation (4.6) can now be written as

Ramembering that No = fn s ) ~ / ~ , we note from the form of equation (6.2) that Mk is a homogeneous function of order (k+ip) in u and s.

Therefore

= u Mk+2u 8 azk + 2 ~ ( k + + ~ ) Mk, by E u b ' ~ theorem ... (6.3) au

Thus

Writing briefly pi for ,ui(Df), we have

Equation (6.4) may now be written in the form :-

This is the fundamental recurrence formula. We notice that ,& is a, homogeneous function of the k-th degree in 8 and u.

S N Bose : The Man and His Wbrk

The general expression for pi can be easily obtained.

From this equation we can immediately write down the coefficients occurring in pi :-

pi.. ...... . I

pi.. ...... . I , p

ph ....... ..I, 2 ( ~ + 2 ) , p@+2),

pi.. ..... ..I, 3@+4), 3(p+2)@+4), p(p+2)@+4) etc.

The general coefficients can be easily proved by induction to be of the form :-

Remembering that u = A2, and s = 1/2t = 2/(np) we finally obtain the general formula for the k-th raw moment-coefficient :-

= Lt. A*. F -k; -k-ip+l; a; - a -+ a, i npAa

The two formulae (7.5) and (7.6) are in agreement with the results obtained by Rajchandra Bose a in equations (8.9) and (8.91) on p. 152 of Sankhyii, Vol. 2, Part 2.

8. It is clear from the preceding considerations that pi(D2)'s, whose recurrence formulae have been calculated, belong to a Gaussian distribution in p-variables; it may be therefore interesting to deduce certain mathematical properties of these functions.

C ~ m p l e t e Moment-coefficients of the LI2-statistic

We start from the recurrence formula

Putting pi = ( 2 ~ ) ~ . $ k k ( ~ / 2 s ) and using X = u / 2 s as the new variables, we have

$k+l(x) = [(k+ ! I P ) + x ] $ ~ ( ~ ) + x $ ; ( x ) . . . (8 .1 )

Multiplying by ez. 2 + + p - l , we have

d eZ. xk+t.-l$ (%) = - [ex k+l dx &+ ' p $ k ( x ) ] ,

By regular descent, since $,,(x) = 1, it follows easily that

d k 6 ~ + + P + ~ ( X ) = ( x 2 z) [ex XY

which may be regarded as a solution of the recurrence formula.

If we put t = 11% = 2s/u, we get,

pi = ( 2 ~ ) ~ . k k ( x ) = ( 2 ~ 1 ) ~ . e-'1' . t ip (---$J '[ell' - t-"1 ... (8 .4 )

which is identical with (8 .6 ) in Rajchandra Bose's paper.3

Starting from the equation ( 8 . 3 ) and denoting by y, the expression exxtp, and d by D , . the operator x2 -, we easily find that y satisfies the differential equation

dx

Operating on both sides of this equation by DL+' and remembering that Dky -- Pk(x) one easily finds that $k(x) satisfies the differential equation

If $ k ( - - ~ ) = ( - I l k . f k ( x ) ,

fk ( x ) satisfies the differential equation


The solutions are easily seen to be polynomials of order k, when k is an integer, and these in carse of (8.6) have been studied by Sonine, Gegenbauer, and others. The differential equations ha.ve also been studied by Whittaker and others, and give rise to solutions of the confluent hypergeometric types.

9. There is an alternative method of arriving a t the differential equation for p', which is here appended as the method is capable of wide generalisation.

The Gaussian distribution function

1 p - - Z @ r . . Y T ) ~ I ~ ~ (2nt)iP

e , where r = 1, 2, 3, ...p

can be easily seen to satisfy the differential equation

where instead of (x,, x,, ... x,), we may also take (y,, y, ... y,) as vsriables.

1 The expression - (2nt)i~ . -Zr(zr-yr)z121 denoted by F also &atisfies the differential

equation

Now, since thi8 differential efiation is linear which-means that if @,, @a, . . . etc. are each a solution of the differential equation then etc. is a solution of the same equation, and since an integral is nothing but a sum, therefore,

1 which consists of a sum of terms like - e -C ( x T - - Y ~ ) ' / ~ ~ satisfying the dif- (2nt)ip

ferential equation and eaoh multiplied by a term like &x,, x, . . . x9) . d$ ax, . . . dxp not depending upon the independent variables the y's, is easily seen to satisfy the equation (9.11).

Therefore

also satisfies the same differential equation.

Complete Moment-coefficients of the p - s t a t i s t i c 157

An orthogonal t~ansformation of (x'~) and (y'$) will not change the integrand; also making a transformation

multiplies the integral by AskE. Hence

or & is a homogeneous function of (R2, t) of the kth degree. Therefore Cl; is a homogeneous solution of the kth degree of (Ra, t) alone of the equation (9.1). Making a ohange of variables to R and angle-coordinates in a p-dimensional space, pi is seen to eatiafy the differential equation

This transformation is seen easily as m i . F = Div. grad F , and when F is a function of R done, the expression follows as the hyper-spherical element perpendicular to R ie proportional to Rp-l. Thus

the differential equation of +;, (s) is easily seen to be

whioh is identical with the equation (8.5)

10. The method sketched above enables one to crrlculate the moment functions even when k is not an integer; actually however, unless k's are integers, pi's will not be expressed by a polynomial but by an infinite series. The differential equation itself determines all the constants except one, which can be determined from simple oonsiderations.


I will conclude my remarks by deducing certain properties of the general differential equation, which will be useful for determining the form of the general distribution function about which I wish to make some remarks in future.

Consider the equation

azP d2F 0;. P = - a2F +-+ ... - aF

i - 2 = ax: ax; ., ax;

Suppose $ is a solution of the differential equation, we then seek a solution of the equation of the form F = $ . X. Then

Now if 1 - - ~ z 3 2 t

@ = @ q W e , then - 8% =-- xr t ' @ ... (10.4)

Therefore x satisfies the differential equation

Suppose F(x,, x2, . .. x,,, t ) is tt solution of the equation

Then @(xi, 5 1 , ... x ~ , t ) = F(xl / t , %,It, ... xJt , - l / t ) ... (10.7)

will satisfy the differential equation (10.6), for putting xi = x,/t; and t' = - l / t , it follows that

a+ d F - = - t ' - 3% ax; *

Complete Moment-coefficients of the tatisti is tic 159

11. This theorem enables us to construct solutions of the fundamental differential equation of the form

where H(x,, x,, . . . xp,t) is a solution of the same differential equation (9 .1 ) .

We have seen that &Re, t ) the moment functions, are solutions of the differential equation (9.1)

where $kk(y) satisfies the differential equation (8.4). Therefore,

XI x, x 1 2' xf+G+ . . .x8. T, ... T9 -7 ) = (- I)'. 9-. h [ ------- 2ta(- l / t ) I = (- (2 / t )k . $k(-y) ... (11.3)

where (- $,(- y ) = f ( y ) which satisfies the equation (8 .6) .

If y = C</2t, and (-1)). $k ( - y ) are the polynomials of Sonine-Gegenbauer, i t follows that

1 (2nt)tP ' e v . ( - 1)k.(2/t)k. $ k k are solutions of (9. I) .

12. A distribution function which satisfies the fundamental equation, and is of a certain type can be written in the form

Certain orthogonal properties of the Sonine polynomials will be useful to recall here. The functions St(x) = (- 1)'. Ct. 'k(-x) satisfies, we have seen the differential equation


which can be written in the self-adjoint form,

from which the orthogonal properties follow a t once.

Taking two polynomials of orders k and kt, we. have m

(k-k') I rz. St. St/ dz = 0 ... (12.3)

Hence if k # k', e4. &-I. &. gk,. & = 0 ... (12.31) 0

since the arbitrary constant in the definition of the polynomial may be so adjusted as to make A = 1, which we may oall the normalised polynomials.

Suppose the distribution funotions to be Hz1, x2 . . . xp, t ) then the distribution function in Ra is obviously

R+dR

&xl, x2 ... q, t ) b 1 b,, ... dxp ... (12.4) R

taken between the hyper-spheres of radii R and ( R + d R ) . It will take the form

A . $(Ra, t ) Rp-I d R = f ( D , t ) . dD when D = RB. ... (12.41)

We shall call p ( D , t) the distribution function, when the number of variables is p.

Then 1 y ( D , t ) D*-l dD = 1 ... (12.6) 0

It is also clear that if y = D/2t, then substituting the expansion of S, (y) which can be easily obtained from the differential equation or from the expression for pi found before, we get

Complete Moment-coefficients of the LI2-statistic

This expression is thus calculable in terms of the moments of the different orders up to k.

Assuming now

and remembering that the above series is so chosen that each term, and therefore the whole series, if convergent, satisfies the differential equation OF = 2aF/at, we have:

From (12.6) since y = D/2t, we get finally

which determines the constants Ak in the distribution formula in terms of the moments. Hence the distribution function can be calculated in terms of the moments. Raj- chandra Bose's distribution function obviously can be put in the form indicated above, the actual calculation would give interesting integral relations involving the Bessel functions.


1. P. C. Mahalanobis : On Teats and Meaeures of Diverpnce : Jour. A&. Soc. Beng. XXVI (1930), 541-588.

P. C. .Mahalanobis : On the Generalized Di&ance in Stetistics : Proc. Nat. Imt. Sc. India, VO~. 11. No. 1 (1936), 49-66.

2. Rejchendra Bose : On the Exact Distribution end Moment-coefticiente of the Dz-stat is ti^ : Sankhyd, Vol. 2, pert 2 (1936), 143-154.

3. Rajchendra Born : On the Distribution of Differences of Mean Valuea in Two 8emplea sad the Dewtion of the DWtatistic : Sankhyb, Vol. 2, Pert 4 (1936), 374.384.

1st June, 1936.

Ramm, Dacca.

Reprinted from Sankhyii - Znd J Stat 2, pp. 386-396, 1936 (Statistical Publishing Society, Calcutta).

S A N K H Y A THE INDIAN JOURNAL OF STATISTICS

Edited by : P. C . MAHALANOBIS

VOL. 3. PART 2. 1937.

ON THE MOMENT-COEFFICIENTS OF THE D~-STATISTIC AND CERTAIN INTEGRAL AND DIFFERENTIAL EQUATIONS

CONNECTED WITH THE MULTIVARIATE NORMAL POPULATION

BY S. N. BOSE,

UNIVKRSITY OF DACCA.

CONTENTS.

1. Algebraic Identities deduced from the Polynomial Form of the Moment-coefficients of the Da-statistic . . . . . . ...

2 Algebraic Identities deduced from the Differential Form of the Moment-Coefficients

3. The Integral Equation connected with the Moment-coefficients and its Solutions

4 The Spherical Form of the Fundamental Differential Equation connected with the Multivariate Normal Population ... . . .

5. The Differential Equation in the Parabolic Form . . . 6. The Differential Equation in the Product Form . . . 7. Purther Reduction of the Parabolic Form to the Polar Form ... 8. Solutions of the Differential Equation in different cases ... 9. On the Series Solotion of the Differential Equation . . .

In a 1)revious paper1 I have considered certain recursion f o r m u l ~ and relations between the moment-coefficie~~ts of the Dz-statistic. Starting with certain further algebraic identities between the moment-coefficients of different orders of the LIZ-statistic, I reach a

number of fundammtal integral and differential equations connected with the multivariate normal distribution. A number of solutions of these equations are discussed in the present paper.

. _ -- --

1 "On the Complete Moillent-Coefficients of the Dl-statistic." Sunkhyb, Vol. '2 (4 ) , 1936, 385-396.


ALGEBRAIC IDENTITIES DEDUCED FROM THE POLYNOMIAL FORM OF THE

MOMENT-COEFFICIENTS OF THE D2-STATISTIC.

1. I start with the identities that exist between the moment-functions of the different orders of the D2-statistic, which can be regarded as the square ~nodulus statistic for a Gaussian distribution in p-variates.

It has been already proved2 that

where x:+$+ ... x; = l2 - and y;+yg+...y; = A2 ( m u )

and pk is a homogeneous function of (A2, dt) of the k-th degree of the form3 :

~ ~ ( 1 2 ) = P k + k(k+p/2- 1) . Pk-l(2t)+. . .

It should be noted here that in (1.2) and in subsequent equations we are writing pk for p; and p for A2 for convenience of printing.

It is easily seen by direct algebraical methods that

So that the various identical relations between p's involving p and t, which will be called the identities of the first type, can be written in the form (Bk)k' = (Bk,)k, or symboli- .tally, (k k') = 0. These identities can be written down by taking any two expressions Bk and Bk, and their existence is self-evident from their form.

Also since Bk's are polynomials in p or in (2t), by taking any three expressions pa = B,, pb = Bb, and /P = B,, we can eliminate (p, p) or (p, t) and thus arrive at identities involving alternatively either p's and t's or p's and p's. These identities can be written as

[abc], = 0, [abc], = 0 etc. ... (1.4)

where the suffix denotes the variable which has been eliminated in the process.

2 Sankhyd, 2(4), 1936, p. 392, equation (9). Sankhyd, 2(4), 1936, p. 389, equation (7.5).

Equations Connected with Multivariate Normal Populations 165

Finally by taking any four of the expressions we can eliminate P, p and t, arrive at relations between y's alone, which can be written as

These idential relations arc interesting, and may be regarded as characteristic of the modulus distribution of the Gaussian type.

Direct algebrsical conlputation will easily verify the following typical relations :-

2pt2-4tp1+(pz-& = 0 ... (1.51)

[1231, = @It+ ~ ~ ( ~ ~ - - , Q + ( P ~ - ~ P ~ P Z + 2~:) ... (i.62)

The expression (1234) = 0 is more complicated; by taking B,, B,, B3, B,, and putting

A2 = ~ f - ~ z , A3 = ~ 3 - 3 ~ 1 ~ 2 + 2 ~ !

I ... (1.61) and A4 = ~ , - 4 ~ # , - 3 ~ ; + 1 2 ~ ? ~ 2 - 6 ~ :

the existence of the following two quadratic equations can be proved,

8ylt2+ 8A2t + A3 = O

I ... (1.62) and A2t2+A,t-A,/12 = 0

from which the bracket (1234) = 0 follows as the t-eliminant of the equations.

2. It is satisfactory to note here that the identities of the above type, which can be arrived a t by laborious algebraical methods, can be deduced much more simply by the following analytical procedure, which furnishes the raison d'dtre of identities of the above types, and especially of the existence of the quadratic equations. This method also allows us to investigate the inverse problem, namely, the nature of the distribution function for the modulus when the identities of the above kind are postulated for the moment-functions.

We start from the differential form of the moment-coefficisnt deduced before4 for the D2-statistic :-

4 Sankhyd, 2(4), 1936, p. 390, equation (8.4).


where t = 2 8 1 ~ according to the present notation.

Then it follows from the fact that pk is a homogeneous function of (28, u) of the k-th degree, that

So that @(t) is a polynomial of - k-th degree in t alone. Also the differential form of $(t) can be transformed into an integral form. Assuming that

where &x) is a function whose properties we shall investigate later, i t follows that

which can be written in the form :-

From this it follows that

Hence by differentiation with regard to t, we have

Remembering the general form of y?k in (2.31) it follom that


This is fundamental relation which allows us to calculate the various differential uoc~fficiwts of $'s in terms of 9 's of different orders. Thus

Since ( $ , t Z ) is a linear function of t it follows that the following differential relations are true.

dL-l$, and generally 1'. '3 + 2kt. ---

a k - ~ ~ l att-1 + W - l ) . F a - = O ... (2.62)

If we now remember that the differential coefficients of 99,'s are expressible in p ' s of different orders, and that $'s we only p's divided by powers of t, the existence of the quadratic equations of the type found by algebraic4 method is evident. This at the same time allows us to writadown the equations with much less labour.

Using equations (2 .51) and equation (2 .2) , we have

Now substituting the values of 9's in terms of p's the above two equations reduce respectively to (2 .65) and (2.66) given below


which can be mitten, by substituting 2s/u for t, in the form

These are the quadratic equations arrived a t by direct elimination and referred to in (1.62)

The fundamental { ) expression alluded to before in (1.5) can also be written in a form involving the differential co-efficients of 's of different orders.

Thus Writing H z -&$-- dk 11.1 ... (2.7)

we have also two relations like

which can be written in the form

Hence the following relation which is equivalent to (1.5) can be arrived a t ss the t-eliminant of the above two quadratio equations :-

There ia however one fundamental relation. The other relations dleduoed should therefore reduoe to the fundamental one doulated above, where the different relations between p's of different orders are taken into account.

THE INTEGRAL EQUATION C O N N ~ T E D WITH THE MOMENT-COR~CIENTS AND m SOLUTIONS.

3. I shall now take up the integral form of $k(L) m

1 a+ e4 . #(z) dk 0 4 (t) = , I ed )(x) dx 0

The form show th& $&)'8 8re moment-funations of a certain x-distribution and


a knowledge of Q(x) will determine the distribution function to which ~ ( t ) ' s can be related. In other words Q ( x ) will determine the 11,'s with which t,hk(t)'s are simply connected. The existence of the nlgehritic iclcntitics between t hc fiLc's follows from two separate hypotheses-(1) the existence of the nbovc. integral rel,tti~)n. and (2) the linear

form of $,(t) . (t)*, We can therefore see what form of @(z) (which for the present discussion can be regarded as arbitruy a t first) is determined by the hypothesis that $,(t) t2 is a linear function of t.

Since

That is,

The assumption of the linear form of @Jt) . t2 implies that

The above relation can be, transformed by integration by parts. Thus

... (3.21) Similarly

The relation (3.13) implies


Thus

This shows that #(x) is a regular solution existing up to infinity of the following differential equation :

if (2) is finite and either A = 1, or &x) = 0, when x = 0.

The differential equation

can be tramformed by the substitution

x=R' /p ... (3.26)

so that

Hence the equation becomes

The different cases for different positive and negative values of A and B can be easily seen to be solvable in terms of the functions of the type J A ( r ) or J x (ir).

When A = p/2, B = 1, the case reduces to that of the Da-statistic. The differential equation then becomes

The solution can be easily seen to be of the form A . R(9-4)/'. I t cp-a)(R)

#x) = A . ( 2 1 / x ) ( ~ - 8 ) / 2 I t (p -ad2dx) ... (3 .32)

Therefore if A is so chosen that


then

Pk Remembering that $k(t) = ---i ... (3.36) ( 2 4

We have = A . 1 ( 2 ~ t x ) ~ e(-lItirlt) . tP12(2dx)(p-2)la . I+1p-1) (24x) dx . . . (3.37)

Assuming (2stx) = r2 as a new variable, then

Q

r= we have pk = A . . e-(A2 + r 2 ) P . 2(~-1)12 . r A

0

Now putting s = Iln, we have

Thus the p's can be regarded as the moment functions for the distribution

-4 n (A2 + 72)

dF = const. n - A(P-21s) rpI2 . I h ( p - 2 ) ( n ~ A ) dr ... (3.44)

which agrees with R. C . Bose's formula.5

5R. C . Boso :"On tho Exact Distribution and Moment co-efficients of the Dz-Statistic". Sankhy& ( 2 ) , 1936, p. 149, equntion (6.4).


4. I shall now discuss the differential equation

which may be regarded izs the fundamental equation for the distribution function of the multivariate normal population in p-variates.

If F is a function of (R, t ) where

then F satisfies the differential equation

THE DIFFERENTIAL EQUATION IN P~BABOLI~ FORM. 5. When F is a function of the form

F ( $ + 4 + . . . + 4 , apl+. . .+agP, t )

by suitably choosing one axis my y,, perpendicular to the hyperplane

a,x,+a#,+ ... a g P = 0

and similar orthogonal axis-system as before, F takes the form

F(R, Y l , t ) or F(y1, $+. . . +Y& t )

The equation (4.1) can be tramformed to the form

where = &+Y:+.-+& ... (6.2)

Further decomposition can be amied on in the same way.

If F is of the form F(xl, xa, R, t ) ; the differential equation can be writfen in the form

where wa = 4+6+ ...+ xi, and so on. ... (5.5)

Equations Connected with Multivariate Normal Populations

THE DIFFERENTIAL EQUATION IN THE PRODUCT FROM.

6. If F(z , , z,, ... x,, t ) can be written as the product of two functions

)1(21i 5 2 , . . . XI, 2) $2 (xri-1, x ~ + ~ , . . . xP, t ) . . . (6 .1)

then the fundamental differential equation (4.1) breaks up a t once to the form

so that the functions ), and 4, satisfy respectively equations of the form

and o ; + ~ . . . ~ 42 - 2. -- - - ~ ( t , a)$, at

Now these equations are transformable to the forms

J

Therefore if L1 and L, are solutions of the equations (6 .4 ) , their product L1 . L2 is a solution of the original equation (4.1). It is easily seen that

satisfies the differential equation

Hence

t P / 2 -

is a solution of the equation in p-variables i.e. of the equation


FURTHER REDUCTION OF THE PARABOLIC FORM TO POLAR FORM

7 . The equation (5.2) can be written in the form

Since

we can introduce in place of r and s,, variables R and 8 define

r = R sin 8 and z, = R COY 8

Then

... (7.11)

d by the equations

... (7.2)

and

dF -- dF cos 8 dF - sin 8. -+ -. - ar d~ R do

The transformed equation in R, 0 becomes

Since xf f xg + o2 = R2

the equation

call Be transfornled by the substitutions

o = R cos 8, x, = R sin t9 cos 4, s2 = R sin O sin $

I a a~ to --(R'-- 1 - a sin^. I d2F R2dR dR + R2 sin 0 88 sin^^ a p . -


Equittion (5.2) could be invest,igeted by assuming a solnt,ion of tho form

so that the equation breaks up in the form

azf (P-1) if hf d2P dP -+T JK-li i = 2:: and --I-(*-?) cot 0 . - do2 do

+AP= O ... (5 .62) 3R2

The cquat8iou (5.4) could be similarly investigated by i~ssunling a solution of the form

so that the equation (5.42) breaks u p into thc thrcc equations

SOLUTIONS OF THE DIFFERENTIAL EQUATICN IN DJFFERENT CASES.

8. We shall first consider the equation of the type (4.1). The existence of a simple solution of the type

is verified by direct substitution.

We shall have occasion to make use of the theorem of Appell and Brill, which has been proved in the case of p-variables earliere, that if H(xl, x,, ... x , t ) is a solution of the differentid equation (4.1)' then

-5 , (xy2t ) e x x L%-- ' ) ... (8.12) Ht(xl, z,. ... . ... rr,, t ) = -- - . H [--L 2

(2n)P/2 , t ' t ' " ' t ' t

is $so a solution.

Therefore in the case of a spherically symmetric solution of the type

H(R2, t) = H(x,, x,, ... xp, 1) ... (8.13)

6 ScmkhyB, 2(4), 1936, p. 393, equation (11.1).


is also a solution.

We therefore t,ransform the equation

by the substitution Rf = y, and the equation becomes

Further putting 2t = T, it beoomes

First consider the equation (4 .3) . If we assume

$(R) satisfies the differential equation

whose solution oan be written in either of the forms

Therefore, from (8 .13) and (8.14) we get two further types of solutions. Carryitlg out the transformation suggested i t is seen that

and

are also solutions of the equation (4 .3) .

and

Equations-Connected with Multivariate Normal Populations

Hence writing h = <2/2, we have as solutions

We shall now seek polynomial types of solution. Stating from the formula. (4 .3 ) and putting

y = R2, T = 2t

we get

Let us put F = Tk . $ ( y l T ) . (8 .33)

Since

... (8 .34)

the equation (8.32) becomes

y . T k - 2 f + ( p / 2 ) . T k - 1 . #' = kTk-1 . #-y . T k - 2 ' . 4' ... (8 .36) >

or dividing by Tk-1 and putting y / T = s ... (8 .36)

This equation has been already discussed in a previous paper.'

It is easily seen that with integer values of k, i t gives polynomial solutions. These solutions are denoted by ,uk(y, T) here. They are identical with the moment-functions discussed at some length in the earlier paper.8 The solutions can be written in the form

The application now of (8 .12) , allows us to deduce solutions of the type

e-R?/2t 1 . - . (t)*/2 Tk S ( s ) , where s = y / T = R2,12t ... (8.42)

7 Sankhyii, 2(4), 1936, p. 390, equation (8.6). s Sankhya, 2(4), 1936 pp. 387-390.

S N Bose : The Man and His Work 178

and

We therefore see that

is also a solution where S,(y, T ) is a homogeneous polynomial in ( y , T ) .

We shall now seek relations between solutions of type (4.3) and (5.3) with type (8.44). We note that since (8.44) is s solution of the fundamental equation (4.3), writing

R2 = y, 2t = T ... (8.45)

we see that

e-vIT L = p,4+* Sk(y /T) satisfies the equation (8.32) . . . (8.46)

that is,

This lesult will be useful later on.

Consider the fundamental equation (8.32) in,y T).

If we write

we can see that

The form of the equation shows that

Ij/. = em F(hy )

is a solution of (8.5%) where F(y) satisfies the equation

Appell's theorem npplied to this form gives us a solution of the form


Remembering that T = 2t, and putting h = A8 .. . (8.57)

the solution reduces to

t-PI2 . e-(R"A?)/2t . J' . (A2R2/4t) ... (8.54)

Remembering again that T = 2t, we see that the solution of (8.54) is of the form t-PI2 . e - R 2 / 2 t . ,-A?/Zt . $(AR/t) ... (8.61)

Now a solution of the a.bove type gives rise to a distribution

where H is a solution of the equation. Therefore

dF = A e - ( R z + A ? ) / 2 t . t-PI2 . $(AR/t) . R(P-2)/2 ... (8.63)

If we compare the result with that in the earlier paper, it is easily seen that the function

$(AR/t) = A . (nhl)-(p-2)/2 . I +(P-2) . (nlh) ... (8.64)

using the notation of R. C. Bose's paper9.

If we remember that n = llt, 1 = R, h = A the identity of the solution can also be easily seen.

By transforming the equation

by the substitulion Z = x2/4, we have

Since IJx satisfies the equation

the solution of the a.bove is easily seen to be

9. I have assumed in an earlier paper the possibility of expansion of solutions or the equation (4.1) as series of the typelo

9 Raj Chandra Bose : "On the Exact Distribution and Moment-coefficients of the Dz-strrtistic" SankhyB, 2(2J, 1936, p. 148, equation (6 .4 ) . 10 SankhyB, 2 (4 ) , 1936, p. 396, equation (12.7) .


where S, denotes A suinn~atioi~ over id1 V L L I U ~ S of k fro111 1 t&m, and Sk denotes a Sonine polynomial of the'k-th order

The justification for the statement will be given here, and I shall, by actually developing the distribution function, arrive a t the same result as before. A distribution function of the type

t-Pl2 . e-(fZ+R2)/2t . F ( p R 2 / 4 T 2 )

where R2 = xf + xg + . . . + xz, necessarily implies that

Assume now that R2/2t = 2, and E2/2t = a ... (9 .13)

The integral then takes the form

Therefore so long as a remains finite

Therefore a constant c can be found such that

so that if we consider the function F(az)-c r Fl(z ) ... (9.18)

we have

Now the Sonine polynomials S&) satisfy the differential equation


or written in the self-adjoint form

We see that e-Z . z(p-2)12 is the dcnsity function for the orthogonal set of polynomials. Hcnce follows the possibility of expansion of a function F,(z) which satisfies the equation

Assuming therefore that F(az ) = Xk Ak . Sk(z ) . .. (9.24)

the co-efficients A, will be (because of the orthogonality of the polynomial functions S, mentioned earlier) found to be given by

Let us suppose that the functions are normalized such that

Then

To calculate A, we at first write az = x ; then ... (9.28)

As noted in Sankhyfi, 2(4), 1836, pp. 394, L = a-(p+a)/2 . e-X/a . S k ( x / a ) . .. (9.32) satisfies the differential equation


Now by successive integration by parts, because the quantities outside the sign fo integration vanish a t both limits, it easily follows that

(C

(3 A d2 i ) ( ) [ ( x F ) - - ( x F ) 2 dx d . . . (9.44) 0

I

we have now

... (9.51) Writing x = az as before, we have

Therefore we have

To detrmine the constant C in the expressicn for A, we observe that

Butwhenaisverysmall C .ak . ea=C.ak . ( l+a+a2 /2+ ...) = C * a k ... (9.64)

Q

Therefore c . = I B, . 8 . e-.Z. z ( ~ - 9 ) / 2 . 8 L (1) . dz .. . (9.65) 0


where Bk is the coefficient of zk in the expansion of F(z) as a power series, as terms of lower order in F(a z) below zk, will contribute zero values because of the orthogonal properties of s~(z)'s

where yk is t,he coefficient of the highest term in the series of normalised Sonnine polynomials, we have

To calculate therefore the coefficients of Bk and yk we consider the differential equation which F(z ) satisfies, that is,

Assuming a series solution F = ZBke zk ... (9.82)

is easily seen that k . (k-1). Bk + a p e k . Bk = Bk-, . .. (9.83)

or k - (k++p-1) - Bk = Bk-, ... (9.84)

Therefore from (9.24), (9.62), (9.66) and (9.85) we have -

P(az) = B, X U P / ~ ) w+ 1) w+ +PI ak ea . 8; ( z ) ... (9.86)

where = S k ( z ) / ~ k

Putting now a = E2/2t, and z = R2/2t

we see that


Therefore the type of solution is of the form

which agrees with the previous resultl1, if we remember the expression

Ak = Ea = & - k(k f i p - 1) ' ~ ~ - ~ ( 2 t ) . . . (9.93)

It follows from the above analysis that the general form of the distribution function in (R, t ) can be assumed to be of the form

where S,, S2, . . . Sk are what can be called Sonine polynomials. The coefficients A,, A,, A, . . . etc. can be calculated in any particular case, when the moments of the modulus distributions are known. The distribution-function in Sankhya12 may thus be seen to fall in the ieneral class, as I have shown by a direct expansion of the expression, and also by identification of the two results.

It is interesting to stydy the general types of solutions of the equation aJ'

0; 6 = 2 ; i ~ for different types of complexity; certain results already arrived a t will

he published in due course.

11 Sankhy6, 2(4), pp. 395-396.

12 Sankhyir, 2(4), 1936, pp. 395.396,

Reprinted from SankhyZ - Znd J Stat 3, pp. 105-124, 1937 (Statistical Publishing Society, Calcutta).

Recent Progress in Nuclear Physics S.N. Bose Professor of Physics, Dacca University.

PHYSICAL science is at present passing through a remarkable phase of radical and revolutionary changes. Time-honoured concepts are crumbling down under the shock of remarkable discoveries. The familiar ideas of force, mass, and energy have undergone remarkable transformations and the dynamical laws which seemed at one time to provide the ultimate basis for the mechanical explanation of the material universe have now been replaced to a large extent by quantum-mechanical rules and probability-calculations. Controversy now rages round the validity of the very principle of causality without which science would have seemed impossible a few years ago. It is therefore not surprising that the old concept of the atom as the ultimate indivisible unit of a chemically simple stuff has been replaced by the modern idea of a complicated structure consisting of a positively charged material core, the nucleus, which itself in ultimate analysis appears to be heterogeneous, enveloped in a cloud of negative electrons, which possess moreor lessdefinite amounts of energy and momentum, determined by quantum- mechanical rules.

The Electron and the Periodic Table

I shall endeavour in this brief discourse to present before you a brief history of the various experiments which have compelled the modern scientists to adopt this structure for the chemical atom in place of the familiar and simple concepts of Lucretius or of Dalton. The beginnings of the change in our point of view may be said to have been initiated about 40 years ago by the discovery of the electron in 1895. The experiments which led to its discovery established at the same time its presence as a universal constituent in all kinds of atoms. The intimate

connexion between electricity and matter being thus established once for all, the subsequent endeavours of the physicists and the chemists have been directed towards explaining the divergent properties of the chemical substances in terms of electricity and electrical forces.

It will be obviously impossible in this brief compass to give an adequate account of all the results so far achieved in this regionwhere physicists and chemists have worked side by side, and I shall therefore confine myself to a bare mention of the principle facts which have led the way to the modern conceptions about the atom. The ancient atomic theory was formulated mainly on the basis of chemical evidence. The analysis of innumerable substances, which either occur as such in nature or are artificially produced in the laboratory, has established the existence of about 92 simple substances, the so-called elements, whose atoms by combining and re-combining among themselves in various proportions have given rise to all the various substances we see around us. Without losing their individual distinctive features, many of these elements show among themselves remarkable similarities in their chemical properties. These have been intensively studied by the chemists, and the main results can be conveniently represented by arranging all the elements in a series of horizontal and vertical rows, in the so-called periodic table of Mendeljieff.

The remarkable feature of this arrangement is that whereas the atomic weights of elements increase steadily as we go down the series, marked similarities in the chemical propkties recur at more or less regular intervals, i.e., as soon as we come to elements which lie in the same vertical column of the rectangular array. The atomic number in the


scheme plays as important or rather a more important role than the atomic weight of elements. No explanation of this regularity can obviously be found in the simple Daltonian theory. Nor was there any prospect from the chemical side alone of arriving at the explanation of the mysterious regularity in the chemical behaviour of elements.

The discovery of the electron however inaugu- rated novel methods of attack for the solution of the puzzle. The presence of electrons carrying negative charges, as constituent of all atoms, as well as the electrical neutrality of the atom as a whole, had brought the interesting question of the electrical structure of the atom to the fore-front. After the discovery of the electron the atom for the physicist could no longer continue to be a simple substance. The problem of its composition, i. e., the distribution of mass and charge inside it, demanded an urgent solution, and speculations were at work regarding its structure even before adequate data were obtained for drawing probable conclusions.

Closely following on the discovery of the electrons, came the discovery of radium and the radioactive elements. Along with the familiar and stable elements which seem to persist unchanged through geological periods of time, the discovery of the phenomenon of radioactivity established the existence of the so-called unstable elements. Though these behave as ordinary elements in all chemical reactions, they are found to disintegrate spontaneously and give birth to new elements of smaller atomic wcights, which in turn explode, regenerate fresh elements, and the series of elements of decreasing atomic weights is continued till the whole process comes to a stop with an element of ordinary stability at the end. During this process of successive disintegration the radioactive elements emit swiftly moving charged particles (the so-called a- or P-radiation). They also emit in general penetrating y-radiations of the type of X-rays. The a-particles were early recognized as the nuclei of helium atoms which carry two units of positive charge whereas the P-rays were found to be swift electrons which move with velocities approaching the velocity of light. Though it was found impossible to control the phenomena of spontaneous disintegration of the elements, their very existence revealcd

the composite nature ofat least the heavy atoms, and made the hypothesis of a structure for all atoms out of comparatively simpler substances a very probable one.

The swiftly moving positively charged ct-particles, emitted during the process of radioactive decay, furnished the physicists with a very convenient weapon for attacking the problem of the constitution of atoms. Lord Rutherford conceived the brilliant idea of sending these swiftly moving charged particles as probing agents inside the atoms, where their mass and enormous kinetic energy would enable them to penetrate far into the mysterious interior, before they would be deviated out of their straight course by the intense Coulombian field of force.

The determination .of the distribution of the scattered particles in various azimuths would, he hoped, enable the physicists to obtain a fairly correct picture of the distribution of mass and charge inside the atom. The first experiments in this direction were carried out in Rutherford's Laboratory at Manchester by Geiger and Marsden, and they at once afforded valuable information regarding the probable constitution of the atoms.

The physicists had already arrived at an estimate of the atomic size from various considerations. The kinetic theory indicated the radius to be about 10-8cm. The ex~eriments of Geiger and Marsden now revealed that the mass of the atom must be regarded as concentrated within a sphere of a much smaller radius, say (lo-'* to 10-'3cm.). This central core was also found to be positively charged, and its magnitude was approximately estimated at half the atomic weight of the element. This experiment of Geiger and Marsden enabled the physicists to form a fairly correct idea about atomic exterior. The planetary atomic model suggested by Rutherford, where a positively charged nucleus controls electrons revolving in orbits controltedby Coulombian forces, gained thus a universal acceptance among the scientists and proved a valuable and fruitful hypothesis. Detailed discussion of the subsequent developments will lead us too far away from our main theme. I shall therefore mention in the briefest of terms the principle achievements which we owe to this model. In 19 13 Bohr showed that the optical

Recent Progress in Nuclear Physics 187

spectra of elements could be explained on the basis of the above model if the electronic constituents of the atoms were supposed to move in definite or- bitsdetermined mainly by electrostatic forces and by subsidiary quantum-laws. The study of the X-ray spectra of elements enabled Moseley at about the same time to settle with certainty the magnitude of the charge of the nucleus and also the number of electrons in outside orbits. The fundamental nature of the atomic number in the Mendeljieffs table received thereby a rational interpretation, and no interpolation, or change of arrangement in the series, could be conceived of at any subsequent period, as the sequence of atoms in the table followed the integral sequences of increase of atomic charges. The idea of the spinning electron gave a rational explanation of the periodicities in the Mendeljieff s table. The electrons were revealed to be grouped in different closed shells round the central core and the chemical properties of the elements could be definitely correlated to the number of the electrons in the outermost incomplete shells. The elements of the same vertical column were found thus to have more or less identical external structure, which explained the similarities observed in their chemical properties.

Structure of the Nucleus

In 1925 came important theoretical developments. The modern wave-mechanics was formulated which enabled the physicists to replace the former tentative calculations by exact mathematical analysis and the different empirical procedures were unified into a single consistent analytical discipline. Though all the questions which have been raised by the study of the physical and chemical behaviour of elements may thus be said to have obtained more or less satisfactory.solutions, the problem of the structure of nucleus had however been scarcely touched till the beginning of the present decade.

It is not difficult to find reason for this delay in the development of nuclear physics. The nuclei of elements lie hidden behind a protective cloud of electrons. The intensity of the Coulombian field also increases very rapidly by about 101° times as we approach from the outside to the neighbourhood

of the nucleus. The bombardment of atoms by swiftly moving electrons had not produced any fruitful results so far, and in order that positively charged particles could overcome the enormous repulsive force and approach the nucleus within a reasonable distance sufficient to produce significant perturbations in the nuclear regions and produce sensible results, swift particles with tremendous velocities appeared at fisrt sight necessary, which were only available in small amounts from radioactive processes that continue yet to be beyond our control. The smallness of the nuclear size makes also the chances of close collision very very remote so that the percentage yield of any definite result by the bombardment of a-particles is very small indeed. Nevertheless since only the chemical properties of elements appeared to be governed by the nuclear charge, an artificial transmutation of elements could be hoped for, if one were able either to push a charged particle in the interior of the nucleus or to bring a charged particle in its immediate proximity, so that the disturbance thus set up might possibly induce spontaneous transmutation of the atom.

In order to ensure stability against the disruptive influences of the Coulombian forces, the presence inside the atom of attractive forces of unknown origin appeared also necessary. These attractive forces are most probably sensible at distances com- parable with the linear dimensions of the nucleus, so that only a careful study of the large deflections of a-particles from single and close encounters with light atoms might be, expected to give some information about the nature of these attractive forces. A successful carrying out of the above programme required the development of a special technique for the study of such atomic encounters. The early method of directly counting the scintillations to estimate the large angle scattering of a-particles as followed by Geiger and Marsden was further im- proved upon by Chadwick and Bieler in the laboratory of Rutherford at Cambridge. A direct measurement of the nuclear charge and an estimate of the nature of the electric forces acting on a-particles in the immediate neighbourhood of light nuclei were rendered possible by the study of scintillations produced by the scattered particle.


The expansion apparatus of Wilson supplied another valuable method of attack. Under suitable circumstances, the tracks of the colliding particles, the nucleus and the a-projectiles, before and after collision, could be directly photographed, and the interchange of momentum and energy between the colliding particles could be directly estimated from the measurement of stereoscopic pictures of such encounters. In case such collisions brought about artificial disintegration, the record of the explosion i n the chamber photograph enabled us in many cases to follow the details of the process unequivo- cally.

The method of directly counting the scattered particles by the scintillations produced on screens has been replaced in recent times by simple, reli- able, and automatic methods of counting the swift particles like a-particles or protons with the help of the Geiger Counter and proportionate amplifiers. The rapidity of advance during*the last few years has been in large part due to the great improvements in the technical methods of attack.

Nuclear Disintegration; the Neutron

Already however in 1919 Rutherford had obtained by the simple scintillation method the first evidencesof artificial disintegration. Nitrogen nucleus bombarded by swiftly moving a-particles appeared to give rise to streams of swiftly moving protons or nuclei of hydrogen, and subsequent work along the same direction had revealed the occurrence of this phenomena of artifical disintegrations during the bombardment of about twelve of the lightest elements.

By studying the phenomena of disintegration of nitrogen in a Wilson Chamber, Blackett was able in 1925 to give a satisfactory account of the details of the process. This nuclear reaction seems to arise out of a capture of the a-particle by the nucleus of nitrogen, whereby an H-particle escapes out of the nucleus and a new atom, an isotope of oxygen, is produced, according to the scheme

Two capital discoveries in recent years have

revolutionized the subject and accelerated thc progress of nuclear physics. Bothe in 1930 observcd that beryllium when bombarded by a-rays from polonium gave rise to a markedly penetrating radiation which appeared to be of the y-ray type. In a subsequent examination of this effect by M and Mme Curie Joliot in Paris and Chadwick in Cam- bridge .in 1932, an important part of this radiation was found to consist of a stream of swift, uncharged particles, called neutrons, which have about the same mass as the proton. As it is uncharged the neutron does not directly i0niz.e the gas in its path, but only reveals its presence indirectly by the recoil of the nucleus with which it collides. And since the transfer of momentum is largest when the mass of the colliding nucleus is about the same as the mass of the neutron, the secondary ionization by recoil, as produced in hydrogen or in paraffin-lined ionization-chambers, is much larger than in an ionization- chamber filled with a heavier gas. This peculiarity of behaviour has served to differentiate the neutron radiation from wave radiations of the y-ray type, which also usually accompany the phenomena of neutron emission. Most of the nuclei of the light atoms alsoemit neutrons when bombarded by swiftly moving a-particles. This strange type of particles is in itself an important agent for effecting artificial transformation of other nuclei, mainly because it is unchargedand as such is not handicapped by the presence of the 'intense Coulombjan forces. It can thus approach and penetrate into the nuclei of even the heavy atoms, and thereby bring out interesting transformations in novel ways about which I shall speak later on.

potential Barrier

The perfection of wave-mechanical methods has induced various theoreti.ca1 workers to apply the new ideas in problems:.of nuclear physics, especially in investigating the collision of charged particles with the atomic nuclei. The simultaneous presence of a Coulombian field of repulsion and an attractive force of unknown origin at short distances inside the nucleus gives rise to what is tech- nically called the potential barrier. The intcnse attractive forces inside the barrier scrve to keep the


charged particles inside the nucleus. Beyond the distance where the height of the barrier rises to a maxinium, the attractive forces cease to be sensible, and the field may be ragarded as repelling charged particles according to the law of inverse squares. Any charged particle wanting topenetrate inside the nucleus will have to surmount the barrier, and according to the classical theory, it will have to have a kinetic energy sufficient to come over the top of the barrier.

Based on arguments similar to the above, one estimated (before the advent of the wave-mechanics) that a minimum acceleration-potential of a few million volts would be necessary to produce sufficient acceleration in particles, before they are able to cross the barrier. The estimate however proved erroneous and excessive.

A noteworthy contribution of the theoretical physicists (recent wave-mechanics) in this subject has been to predict a small but appreciable probability for penetration of protons across the barrier of light elements, even when they have energies of the order of a few hundred thousand e.volts. The prediction seemed to bring artificial transmutation on a large scale within the range of possibilities, and this theoretical conclusion was tested experimentally by Cockroft and Walton in 1932 who had been able to generate a proton stream of about 100,000 volts, in a vacuum tube, by step-wise acceleration. Their experiments at once met with complete success and the evidence of disintegration of the lithium nucleus subsequent to the capture of a proton was obtained by bright scintillations produced on the screen by the a-particles that were generated by disintegration. This initial success of the Cambridge physicists has been followed up by other workers in Germany, France, and America, and it has become clear that considerable progress in artificial disintegration can be achieved by bombarding elements with swiftly moving protons and other nuclei accelerated suitably by application of high voltages. Investigators of nuclear physics have concentrated their energies on the production of suitable high voltages, and notable success in this direction has been achieved in recent years. Three different methods have been mainly followed in producing.the necessary high voltage : firstly, the original method

of Cockroft and Walton, which consists in acceler- ating the particles in the vacuum tubes in stages with suitably insulated transformers; secondly, the method of Lawrence, who has developed a peculiar method of multiple acceleration of ions in a synchronized magnetic and electric alternating field; and thirdly, the electrostatic method of Van de Graff. Consider- able progress has been achieved by the application of all the three methods and a lot of interesting results have ben obtained by the bombardment of atomic nuclei, by suitably accelerated projectiles. I have already mentioned before the production of neutrons from the light elements by the bombardment of a-particles. Results of great significance were obtained as soon as these particles were used by workers in nuclear physics. In curious contrast with the swiftly moving a-particles and the artificially accelerated proton streams, and other corpuscular rays, the capacity of the neutron to produce artifical transformations increases, in most cases, with the diminution of its velocity. Doubtless this is connected with the fact that its small velocity enables the particle to stay longer in the immediate neighbourhood of the nucleus and thus to bring about more far-reaching changes.

Induced Radioactivity

It has been established by the work of Fermi and other members of the Italian school that a few collisions of neutrons with hydrogen nuclei (present in either a free or a combined state) are sufficient to establish a sort of thermal equilibrium, so that the neutrons on the average attain, after a few collisions, the averave velocity of hydrogen particles at room-temperature. Whereas the transmutation of elements to stable varieties of other elements had been previously noticed or conjectured, during the bombardment by a-particles, or protons, Curie and Joliot discovered in 1933, that unstable and hitherto unknown isotopes of light elements are produced by the bombardment of light nuclei with a-particles. These unstable nuclei subsequently break up in the same way as ordinary radioactive elements, and phenomena of induced radioactivity are found to obey the same laws, and decay in the same characteristic way as the natural activities of the well-


known radio-elements. The identity of these artificial bodies can also be often established by purely chemical methods. In their pioneer work Curie and Joliot discovered that unstable isotopes of nitrogen, silicon, and phosphorus are formed by the bombardment of Be, Al, and Mg: these however emit, curiously enough, positive electrons though the phenomena of disintegration proceeds like the familiar p- ray disintegration of radioactive elements. The discovery that new radioactive elements can be produced by artificial transmutation has given a tremendous impetus to the study of the nuclear reactions. The taskof following such nuclear changes is a very much easier one, as sensitive physical apparatus like the Geiger Counter can be utilized to detect and measure such changes. This pioneer work of Curie and Joliot has been greatly extended by Fermi and other workers in Italy. They have observed that slow-moving neutrons are in most cases quite effective in producing similar changes and of generating radioactive isotopes. Extensive work in this line has been done and a large number of new radioactive atoms have been discovered. These however almost always emit P -particles, i.e., ordinary negative electrons instead of positrons.

Deuteron-the Isotope of Hydrogen

With the discovery of deuterons, the isotope of hydrogen, a new kind of corpuscle has been utilized for bombardment of atomic nuclei. Acclerated deuteron streams have been utilized both in England and America and they have proved very much more effective as agents for transmutation than the original proton rays of Cockcroft and Walton. Bombard- ment by deuterons also produces radioactive bodies and this method has been increasingly used in recent times to study the phenomena of induced radioactivity. The earlier methods of production of new radioactive bodies had to utilize a natural radioactive source for the supply of the suitable bombarding agents; this had necessarily limited the amount of the yield even when the process of canalization was utilized for diminishing the velocity of the neutrons and thereby to bring about an enormous increase of output. The yield of radioactive stuff by the bombardment of deuterons has been naturally

very much greater. For example, in a day's exposure a yield of radioactive isotope of sodium has been reported to have been obtained by Livingstone whose activity equals that of 1 gramme of radium. Deu- teron bombardmeni has therefore a great future both in the therapeutical application of radioactivity to medical research as well as in the investigation of nuclear.problems.

Whereas bombardment of corpuscles has been almost always utlized forproducing nuclearchanges, results similar to photo-ionization have been obtained by Chadwick and Goldhaber. By utilizing hard-y-radiation from ThC these investigators have been'able to decompose the hydrogen isotope into neutron and proton. The liberated neutron can be detected by its ability to produce induced radioactivity in suitable elements, or by a properly con- ducted ionization measurement. By hard X-rays emitted in tubes run at more that 1.5 million-volts pressures, the workers in Berlin have been able to eject neutron streams from beryllium, whose presence has been similarly demonstrated by the generation of radio-iodine in ethyl iodide. These preliminary results have great theoretical significance; Chadwick has been able from a tentative determination of the thresh-hold value of the frequency of the y-rays necessary to decompose deuterons, determine the mass of the neutron as well as the strength of the binding of the two fundamental particles.

Conclusion

In this lecture I have attempted to give a rapid review of the principle results obtained in the field of nuclear physics in recent times. I shall conclude my remarks by mentioning two remarkable results that have been obtained by the bombardment of thorium and uranium, the two heaviest of the elements with neutrons. I have already observed that the production of new radioactive bodies could, in many cases, be .proved by chemical methods; this serves at the same time to determine'the chemical properties of the new substance, and its position in the periodic table. By the bombardment of uranium, Fermi originally reported the production of new P- ray emitting elements whose chemical behaviour seems to point to their positions beyond uranium


itself in the Mendeljieffs table. This has been cor- roborated by Meitner. According to these workers the table of elements is artificially extended by this process beyond uranium itself-according to the scheme

P uE8+ nb* u;i9- Eka Rhenium E R ; ~

Eka ~ s ~ ~ ~ - - - - + ~ k a ~r~~~ 94 95

a truly remarkable result.

By a study of the product of the disintegration of thorium by neutron bombardment Curie has established the existence of a new series of radioactive elements whose mass numbers are in the form 4n + I . This analogous series of radioactive bodies is not known in nature, and its discovery can be regarded as completing our ideas of radioactive disintegration of heavy elements.

It will be evident from what I have reported above that enormous experimental materials have accumulated in the field of nuclear physics within recent years. Sufficient materials are now available for the theoretical physicists to speculate about the process of atom-building from elementary particles,

and the preliminary work in this direction has already begun. The neutron and the proton seem to be the two exclusive constituents of all atomic nuclei. The a-particles can themselves be looked upan us composite bodies, built up again from neutrons and protons.

Though the idea of the elementary atom has undergone revolutionary changes in recent years, in a certain way the progress achieved has been satisfactory, as i t has simplified the number of ultimate and fundamental particles to only two, instead+f the 92 elements of the older atomic theory. This idea of the evolution of the material world from comparatively few primordial stuffs is not however new. Here, as in other fields of physics, old ideas have returned, renovated in a new garb and clothed with more significance. The quantum theory of photons has to a certain extent resuscitated the ancient corpuscular theory of Newton. The recent developments of the nuclear physics have brought back the old Proutian hypothesis, of the evolution of all elements from one or rather two primordial stuffs- the proton and the neutron.*

*Delivered as the Adharchandra Memorial Lecture at the Calcutta University Science College on the 21st December 1936.

Reprinted from Sci & Cult 2, pp. 473-479,1937 (The Indian Press Ltd, Calcutta).

Anomalous Dielectric Constant of Artificial Ionosphere

In a rqcent communication in Nature, Mitra and Roy 1 have pointed out an interesting feature in the ionospheric dispersion formula. They have shown that the formula can yield values of the dielectric constant of an ionized medium greater, equal to, or less than unity depending on the degree of ionization, collisional frequency and the exciting wave frequency. They therefore hold that the value of dielectric constant of an ionized gas greater than unity which has been recorded by many investigators for comparatively large ionizations is only an outcome of the complete dispersiori formula. The object of this note is twofold. Taking the ionospheric dispersion formula we shallfirst deduce the conditions under which the dielectric constant may assume values greater or less than unity and secondly we shall show that in the experiments where anomalous values of the dielectric constant of an ionized gas have hitherto been obtained, the experimental conditions are such that the dispersion formula alone cannot explain the anomaly.

The dispersion formula is given by

mpL mP" where a = -- and p=-

4 n ~ e ' 4 x ~ e '

Separating the real and imaginary parts, we have

2ck B and - p = ..... P d+g2

e 2 k 2 la1 Thus - - p = - - 1 so that

P d + p 2

> C2k2 ' la1 - p2 = 1 according as 7 - < P < d+P2

which is equivalent to the condition :

p2c2kZ - 2 but as from (2)

p2 4 ( d + p 2 )

the condition reduces to -

Now the left-hand side is always a proper fraction;

therefare, if 4 la1 > 1, i.a la1 > .25 we have

then always

If however la1 < -25. p2 > 1 will require the further condition

Calling & = 2 or rn Po 7

2

we have 4 la1 = and we can ray that if

p,'

Anomalous Dielectric Corzstant of Artificial Ionosphere 193

p > po = 2'8 x 10' f i , pis always less than unity,

whereas if p < 2.8 x 10' @, or la( <'25,

for p2 > 1, we must have

i..e., the collisional frequency must exceed the value

Thus the equations.show that even when p < 2'8 x

104 n, there exists acritical collisional frequency which has to be executed if p is to be greater than unity.

air in a discharge tube has been found to be less than unity when the tube current is small. As the ionization is increased by increasing the tube current, the dielectric constant at first decreases and then increases gradually and even becomes greater than unity. The maximum electron density N in our discharge tube has been within 107. Taking N=lO7,

p=5 x 108, we get la1 = 7'8. This is much greater

than '25 so that the dielectric constant must be less

than unity. Even when N=108, la1 ='78 and the

dielectric constant should be theoretically less than unity.

In Fig. I are plotted the calculated values of the dielectric constants against h2 for some definite values of N and v . Curve (2) which corresponds to Mitra and Banerjee's3 experimental conditions shows a turning point at hof theordergm (h2is of the order 8x105 sq.cm.) which is very different from the experimental value h of the order 6m (h* is of the order 3'6x105 sq.cm.). In the experiments of Ali Imam and one4 of us, which have been recently repeated by a different method, the turning point appears unmistakably at h of the order 4m (h2 is of the order 1 '6x105 sq.cm.). Curve (1) which approximately corresponds to our experimental conditions shows however the turning point at h of the order 12m (A* is of the order 14x105 sq.cm.). It appears extremely doubtful if the disposition of the experimental apparatus can account for such wide dis- crepancy. It is likely that there must be some other explanation of these experimental results.

Physics Department, Dacca University. S.N. Bose,. 13.1 1.37 S.R. Khastgir.

SQUARE OF WAVELENGTH (SQ. CM.) 1. Mitra & Roy, Nature, 140, 586, 1937.

Fig. I 2. Appleton & Childs, Phil, Mag. 10, Dec. 1930. In the experiments recently performed in this 3. Mitra & Banerjee, Nature, 136, 5 12, 1935.

laboratory, we have obtained results similar to those of Appleton and Childs2. Using a wavelength of 4. Imam & Khastgir, Phil. Mag. May 1937 and

about 3'8 metres the dielectric constant of ionized Ind. Jour. of Physics. 10, Part I , Feb. 1937.

Reprinted from Sci & Cult 3, pp. 336-337,1937 (The Indian Press Ltd, Calcutta).

ON THE TOTAL REFLECTION OF ELECTROMAGNETIC WAVES IN THE IONOSPHERE

BY S. N. BOSE, DACCA UNIVERSITY.

(Received for publication, April 5, 1938)

The conditions under which the ele~tromagnetic waves get totally reflected in the upper atmosphere appear still worth a closer investigation. The nsual procedure has been to attribute to the medium a refractive index which is calculated according to the classical method of Lorentz from the mechanical equations of motion of electrons. .In an absorbing medium, this refractive index is a complex quantity, and depends not only on the properties of the medium but also on the frequency of

the waves : progressive waves through the ionosphere are damped, the co-efficie~lt of damping will also enter into the expression of the refractive index. The usud Appleton-Hartree condition for total reflexion has been deduced by neglecting the damping, as a &st approximation, and by putting the refractive index equal to zero. Discussion of the conditions when the damping is not negligible, gets very complicated indeed. Certain workers in Allahabad have suggested the further condition that the group-velocity of the wave-train is zero when total reflection takes place. This is a reasonable hypothesis, but the method by which the group-velocity is calcul~ted neglects damping, an essential factor in the physical process. It appears also that the calculation of the goup-velocity of waves in an absorbing medium is not atnen'~bk to the ordinary methods unless the damping is negligible, an approximation which, as has been already pointed out, does not seem to agree closely with experience. I t seems therefore necessary to change the method of analysis and to take, instead of the equations of Maxwell with a complex refractive-index, the microscopic equ a t ' lonlj of Lorentz as basis for oalculation. The method sketched here is a general one suitable for problems of refractive-index of material media as well as for discussion of propagation of waves in the ionosphere. It seems further amenable to modifications to suit the requirements of modern wave mechanics.

We take the familiar equations of Lorentz

div E = p , and div H = 0.

The density and the convection current, p and pV, can be regarded as decom- posable into two parts p,, and p- and (pV)+ and (pV)-, respectively, which thus takes

Reflection of Electromagnetic Waves 195

account of the corpuscular distribution of positive and negative charges. A wave disturbanoe in the medium can be considered by putting

V+ = V,O+ W+eS and V- = V!+ W,eS, where the elements with 0 indices correspond to the undisturbed state of the medium. The convection current, and the electric-density become on this assumption equal to

pV = p+v++p-v- = (pV),+8eS

where fi = [p+~ w++~+v,o+~!w-+q-~!) and 00= q++q-.

Thus the components to,, O,, 0,) of the vector 8 correspond to the amplitudes of the fluctuating part of the convection current, and 0,, similarly to the fluctuating part of the electric density due to the disturbance.

The process is a familiar one : When S is a linear function of x, y, z, t, with a complex factor '2n1, as is generally chosen for disoussing monochromatic wave- propagation, 2n1 S represents the phase of the disturbance and the surfaces for various values of S, represent the totality of all wave fronts as function of (x, y, z, t). In order to tackle the more general case of a damped wave-train we shall not a t first put any limitation on the form of 8 but shall assume it to be any function whatever, whose differential co-efficienta may also have complex values. Removing the contributions on the both side of the equation by the various quantities corresponding to the undisturbed state, we get

div (EeS) = B0eff, and div (Hes) = 0. We can also verify easily that

div [Beff] = [div E +(grad B.E)]eff

curl [EeS] = [curl E +grad 8 x mes

eta.

Under the usual condition8 of propagation 1, 8, representing the amplitudes of the electric and magnetio waves, are slowly varying quantities whose differential co-efficients with regard to time and space co-ordinates can be neglected in comparison with the rapid variations of the function S which represents the ,phase of the disturbance. We shall therefore put the differential co-efficients of E, H equal to zero,

196 S N Bose : T h e Man and His Work

and thus get finally the equations reduced to the following form :-

(grad S E) = 8, (pad 8 - H ) = 0

These can be regarded as the characteristic set of the wave-equations of Lorentz,

as as as 'as are taken as unknown, and since there are eight equations, if - - ax 9 ay a~ at certain compatibility conditions have further to be satisfied which will give us the conditions under which wave-propagation can take place. We proceed to calculate these conditions by the usual method of vector-analysis. We have only to remember that, as the components of vectors may have complex-values, the geometrical interpretation of the various quantities may not be so immediate as in the usual cases : also a relation A2 = 0, does not mean that the vector vanishes, as the components of the vector A,, A,, A,, need not be zero, when A:+Ai+Aq = 0.

We shall call such vectors as singular, and in the problem that we have got before us, considerations of singular-vectors become important.

A few remarks regarding the geometrical interpretation of a singular vector may not be out-of-place; regarding the components of a singular vector A, as de- composable into 1ea1 and imaginary parts, we can alway%i ahoose a real direction ( A , p, V )

which is perpendicular to a complex vector, viz., (A,+AZ L, ...)

z.e., such that AAzf~AV+;vAz = 0

i.e., an imaginary vector A, may be regarded as defining a real plane.

If cohordinates are chosen so as to represent this plane as the plane of (my), a singular vector A, i.e., whose A2 = 0, can be represented as

(A, LA, 0) where A, may be a complex quantity, i.e., A is of the form[iA+i,u,

-p+ih, 01. Any vector perpendicular to a singular vector, will have in this co- ordinate system, the general form (P, LP, Q) where P and Q may be complex quantities. A non-singular vector, C perpendicular to A, will have C2 + 0, i.e., Q # 0; a singular veotor perpendicular to A, will also necessarily have Q = 0, Le., if A and B are two singular vectors! both conditions (A-B) = 0, and ( A x B) = 0, may be satisfied at the


same time and all singular vectors perpendicular to one another may be regarded as lying in one real plane.

If A and B are two perpendicular vectors of which one is singular while the other is not, any vector which is perpeinr!diculay to both A and B, must necessarily be singular, and will lie in the plane of A. This proposition follows immediately by d t i n g out the general form of A and B in the manner explained above, and trying to write down the general form of C which satisfies the orthogonality condition. These remarks will be useful in discussing the singular case of total reflection, as- we shall see later on. The other two relations in'vector analysis which we shall have to use pretty frequently are

A. ( B x C ) = B.(C x A ) = -B. ( A x C ) = etc. . .. ( I )

and A x ( B x C ) = B(AC) - C(AB). . . . (11)

These have unrestricted validity even when the components of vectors have complex values, as they are formtd identities, involving only re-arrangement of terms.

We shall at first deduce certain general conclusions from our equations

(grad 8 .E) = 8, ... (3)

(grad 8 . H ) = 0. . . . (4)

Scalar multiplication with grad S of (1) gives, taking account of (3),

. .. (A)

a relation which expresses the conservation of charge in the disturbed state of the medium.

Vector multiplication by grad S of ( 2 ) gives


taking account of (1) we get

1 dS 2 19 +8,, grad S = 0. [ a ( x ) grad^)^]^+^^ . . .

Scalar multiplication of (2) with E gives (EH) = 0 . . . ( C )

Scalar multiplication (1) with H gives ( 8 H ) = 0 . .. (D)

Relations (C), (D) and (4) show that the vectors E, 0, and grad 8, are all in one as

plane perpendicular to H ; also (1) shows that grad S is perpendicular to - E+8. at dD

This vector can be regarded as representing the rate of change of induction, or at

of Maxwell's equation; thus H and D can be regarded as lying in the plane .perpendicular to grad S while E and 8 separately do not lie in it.

The arrangement of the various vectors thus agrees with our usual ideas about electromagnetic waves.

ENERGY AND MOMENTUM RELATIONS IN THE WAVE FIELD

Scalar multiplication of (1) and (2) by E and H respectively give

1 as h T ~~+(* '=E . (g radSxH)=- -g radS . [ExH] C J ... (E)

[ as --Hz = - grad S. [B x-HI ; c dt ... ( F )

addition and subtraction of (E) and (F) give two further important relations

1 a s [ E ~ + H ~ J (Ee) 6 3z- 2 - - grad 8. [EX HI +2c-

1 as -- (Ea-Ha) =-- (Be) . at C

Also vector multiplication of (1), and (2) by H and E similarly gives

. .. ( G )

. .. (K)

Reflection of Electromagnetic Waves

also E ( E x ~ ) + ~ l ~ r a d S-& (grad S.E) = 0 c at

or - - as ( E x l l ) + E Z g r a d S = 6.E. c at-

Subtracting (L) from (I<) we have

grad S (B2- E2) = - I BoE 4- --c L

Multiplying (K) and (L) by grad S we have the further relation

but I a S

grsd S. ( O X H ) = - 6. (grad SX H)=- - (on+ -- (EB) ) C at /

1 - O2 1 a s -grad c S. (Ox H) =--I-+- c c2 3t ( ~ o ) ]

These two relations thus become

From these we get

02 - - - 4 + 1 (E0)-(grad S)2(Hz- E2) = 0, c2 c2 dt

or remembering (H) we see that

1 / d S 2 8 2

[> :. ar 1 - (grad S)z (Hz--E2) = S 0: -$.

Also (grad S)2H2 = (grad S.H)2+(grad S x

. . . (T,)

. . . (RI)

. . . (P)


we get

1 a s -- 1 a s a s c at

grad S . ( B x H)= -- -- [ EZal +(I&?) ] c2 at

which can be written in the form

This equation could have been immediately deduced from (B) by taking scalar

multiplication with E.

Equation (B) :

gives us generally the mutual disposition of E, 8 and grad S when wave-propaption is possible, whereas the equation (Q) written in the form

may be regarded as the eeuation for the refractive index, determining the velocity of propagation in any direction prescribed by g a d S.

A surface S(x, y, z, t) = C may be regarded as moving perpendicular to itself with a velocity given by

C the refractive index can be defined as - = p= W

I grad I , agreeing wit11 our usual 1 a s - - c at

definition, which however can be complex in the general c a w

Introducing p, equation (Q) can be written in the form


Early in the paper we have expressed 0, in terms of motion of tlic positive and negative charges, namely,

These involve in general both the original distribution of thc: charges, tlioir velocities as well as the amplitudes of velocities, and densities of positive and ncgn- tive electricity clue to the disturbance in the medium.

From general considerations we can express the relation of 0, and E in tho following way.

We have seen that during propagation of disturbanoe both O and E lie in the plane perpendicular to H. Therefore, 8 can be decomposed into two components, one along E and another perpendicular to E, in this plane, i.e., in the direction of ( E x H).

8 thus can be put as equal to a E + p

also from equations of wave-propagation

I ) x E = cH (grad S E) or p / c a = 0g/E8.

Writing therefore

we see that

so that the refractive-index equation can be expressed in terms of a and B in the form


As t1.e vectors I? nnd li are perpendicular to one motlier it follows that rand are two

oo-efficients which stand ill very ainlple mlution to the total po1,zris:rtion current;

a nncl /I however are not constnnts hut may involw grntlS its well as dB . &, 2.e.,

111a-y lw fi~notiol~ of tlircbctio~i mtl frequency of the waveq.

COSDTTTON FOR TOTAL REFLECTION

I t is well-known that for electro-magnetic waves in vacuum we hare E2-H2 = 0

as well as (E . 11) = 0. For waves in inaterial media however E2-H2 # 0, but (Eli) = 0 remains still valid. The rntio of the electric and magnetic vectors depends generally upon the properties of the medium as well as on the frequency of the wave and we can. express this fact by the relations (H) and (M)

which are

The ratio of the amplitudes of the vectors, Ha/EB, can therefore be expressed by the following relations :

Depending upon the property of the medium and the direction of disposition of the d S d S vector E , F ) will generdly vary with - as well as with grad 8. If, however, -

~4 at at

(3s tends to such a value that - 4- 7. =0, it follows that Ha = 0 (1 #'O), indepen- d t

dent of the amplitude of the wave traversing the medium; the dieturbance in

such a case loses its wave-character,,so that we can say the wave cannot penetrate beyond that region and gets totally reflected. We shall therefore take this condition Ha -- Be - 0 or an equivalent condition as characterising the condition of total reflection


and we can also correlate it with another physical idea.

We have also proved the following relation (F)

and also

EO -grad S . ( E X H ) .

C

grad S.(E x H ) may thus be regarded as proportional to the flow of energy ~ L C ~ O S S t l ~ surface S = constant, so that grad #.(EX H ) = 0 may be interprctecl to mcm th~rttlio flow of energy across the wave-front vanishes. This ngrecs with our usunl itlw of total-reflection a t a boundary of two media, where the energy flows along the boundnry interface in the limiting case.

We have also seen that

1 grad SI = , ~ r = the refractive indes of the mccli~uin.

1 as

as 'HZ = (grad S x E)2 = E2(grad S)2-(grad S.E)? From (2) .we have - c2 (3d

we have

Hz - (grad S)2 (grad S.E)2 - grad S.[E x HI - - H.(grad S x E l EZ 1 a s 2- 1 as 2 = - 1 as

-. ---E2 1 as - a i r ) E & ) c at - c at --B2

Ha I n case (grad SUE) = 0, we see that the condition -= 0 is equivalent to the E 2

conditionp2 = 0, as has been assumed by Appleton to characterise the case of total refleotion. If we now examine the condition (B) which gives the relation between the vectors E , 8 and grad S, the condition 8 = 0 will mean that the wave is propagated under such conditions that the vector E is paxallel to 8.

Multiplying (1) with E , we havo

E x 8 = cEx(grad S x H )

= - cH(grad S - E),

so that (grad S E ) = 0 will mean E X 8 = 0, i.e., E is paralled t o 8.


\Vc thus see that if the wave is such that E is 11 to 19 the condition of total reflection JI = 0, or grad S.(E x H) = O is equivalent to theassumption ,u = 0; in the general case of wave propagation when E is not parallel to 0 we would expect on the other lmd thc condit,ion H2 = 0 as giving a new result.

H" 0, or grad S.(E x H) = 0, may be regarded as equivalent to the assumption tllibt tlic group-velocity vanishes. The condition of total-reflection postulated abow is obviously suitable for a train of damped waves, because om analysis is general and the conditiou of non-penetrability has been deduced independent of all ideas of frequency.

W7e shall now apply t,hc results of our analysis in the particular case of the ionosphere.'

The contribution of the positive charges, as well as of the bound electrons in 0, may be easily seen to be negligible in comparison with the cont~ibntion of the free electrons.

We can assume for the ionosphere

where these quantities refer only to the free-electrons present. If we average over s volume containing a large number of electrons, but small colnpsred with the wavelength of the disturhmce,

- p; = Ne, O = N e w ;

the average value of the second term rgv,, may be regarded as zero as the initial velocities of the charges may be regarded as distributed chaotically in dill directions.

Writing the equation of motion of a free electron after Eorentz as

mg+gz = c [E ,+~~-V%Y~ , , etc. C

where (h,, h,, h,) may be regarded as the components of s steady magnetic force, we have, dividing by m and multiplying by p = Ne, and averaging, the following equation

- where (0, = Nev,, etc ...).

e B Remembering that for an oscillatory disturbance 8 = 8 -- as before idso that the at


other terms in the equation varies a t eS, the equation can be written as

cancelling out es. = -8 is usually called the collisioil frequency of the electrons. \Ye rn

(E'4 @ shall use this equation (Z) to calculate -- ~2 or^'

Cme I : This can be regarded as decoinposable into two subcases.

(a) When '2 = 0, ix., R is parallel to the external mynet,ic field, equation C1/1,

(Z) shows that 0 is then parallel to E

Tho equation for determining the critical frequency for total reflection becomes

This determines the dependence of the critical frequency on the ionic density; the train totally reflected is therefore of the form

or u:; = Ne2/m- v2/4

and the damping co-efficient is v/2.

Case I1 : When 8 (1 E and 8% = 0

that is the vector 8 is singular, d2 = 0 but 8 # 0

a vector (8 x$) will represent s vector parallel to 8, when 6 is non-singular and

perpendicular to 8 according to our former remarks about singrilar vectors. Choosing the dire~t~ion of h, as the direct,ion of 8, we can write the relation


in the form

6' Ne2/m - becomes equal to E

cm

according as 0 and E are of the form (A, FLA, 0).

The condition of total lcfiection now becomtts, from (B),

calling

we have

Solving the above equation, we find that the two reflected waves are circularly polarised and the critical frequencies are

eh eh So that when we have the critical frequency =- i.e., w r =--., one of tho

CNZ ' 2cm circularly polarised wave will be suppressed, and the other will be reflected with un-

Ne2 diminished intensity : as this will necessarily mean-- = $14 the reflection of planc-

m

Reflection of Electromag~tic Waves 207

polarised wave, discussed above in case I, will not occur as the condition of total Ne2 - .v2/4. reflection gives a, = 0, if -- m

These two cases correspond to. the cases discussed by Appleton by putting p = 0 and the result agrees with the usual formula, when the collision-frequency is neglected. Case 111. When8#E:

We shall now caluulate the value of ( E ' i n the general owe. E2

w e have

Scalar multiplication with E gives

also vector multiplication with E gives

eh Scalar multiplication by - of the above relation gives cm

Again, scalar multiplication of the original equation with *gives cm

Combining (a), (b) and (c) we can easily deduce


when E and O are uon-singular vectors.

Now if we consider the oase of total reflection as defined by grad S . [E x H ]

Now a h grad S [Ex HI can be written in the form H(grad S x E), we see that

grad SIP

and

1 eh %%- (& )*cot3 C if grad S is s v e e h whose compo-

nents have real ratios.

as ( ~ e ) The relation- +- == 0 becomes dl E%

If the disturbance is plane-polarised and the collision frequency is ernall we can put

where a is the angle between the direction of propagation and the magnetic lines. as The formula b e m e s , putting - = i a, at


agreeing with the formula supposed to be derived by putting group-velocity = zero by Rajpai and Mathur.'

We shall have to remember however that E may be a complex-vector in the generd case, any complex vector E = [E,+iE,, eto.], may be regarded as perpendicular to a real direction, and choosing this direotion as the direction of 2, we oan choose as components of the complex veotor, (E,, L E,, 0) where z, and y are certain special directions and the external field is regarded as having components (h,, h,, hz) in these special direotions.

(E' ) then beoomes equal to -2- (E>,+r E&,) cm

"

The frequency equation for the elliptically polariaed beams would be

when the collision frequency is neglected this similarly becomes

This shows that the two cases will be either h, = 0. or h , = 0

We have tiherefore


when h is 1_ to the major axis,

when h is 1_ to the minor axis.

THE PROPAGATION OF THE DISTURBANCE

We shall analyse now the relation (B), i.e.,

which may be regarded as prescribing the relation between the three vectors E, 6 and grad 8, when propagation takes place.

For this purpose, we shall have to consider the relation (A) which expresses the conservation of charge, it?.,

1 - as -- c j at

@+(6 grad S ) ] = 0,

and the relation (Z), i.e.,

We can utilise (A) and (Z) to eliminate E and 6,'from (B) and thus get

or by re-erranging terms,

as n as 2 eh - N"' grad S (8. grad 8). ( R) -r[&-) - ( @ ' s ) ] ( s x ~ ) -

This vector equation is equivalent to three homogeneous linear equations in (O,, 6,, 8,),


whose co-efficients involve the differential co-efficients of S and the collision-frequency; when the relation is satisfied for non-zero values of (O,, B,, 0,) the discriminant of the simultaneous equation must vanish. This will give a relation between the differential co-efficients of 8 alone, where the characteristic numbers of the ionospheric layer, i.e., N, h and v, will also be involved, and may be regarded as giving the equation of the characteristic sur'face, corresponding to the Ei-konal equation of Hamilton for a material medium

or equation for vacuum.

It is better for perspicacity to introduce certain abbreviations a t this stage; we shall put

where

1 a s 2 as as Ne2 [a ( ai- ) -@ad's ] ( s + v ) +;;;;, ( %: )'is put

The vector equation (R) is thus written

Scalar multiplication with 0 gives

Nea L(8)82 =-(8. grad S)2,

m

so that (grad S 0 ) = 0, either when f)2 = 0 or when L(S) = 0, so that the vector

equation becomes simplified, in both of these cases.

E I 1 0

Case I. When L(S) = 0, 8% # 0.


The vector equation (R) is satisfied only when

eh as ( exE; , ) = o , or s(s)=o, or- = o . at

3s We reject = 0, as we are considering wave propag8tion. We can easily see that

t g(S) = 0 is a singular case, which is not relevant to the problem before us, il.5 y(S) = 0 will lead to the vanishing of the vector 8.

0 is parallel to h, the external magnetio field.

The characteristic equation thua becornea

or putting

which agrees with the ordinary expression for the refraotive index for plane polarised ordinary ray. The singular solution of the equation (1) when p(S) and Igrd S I = O givea the case of total reflwtion.

Case 11. When 8% = 0

The v e o b equation (R) ean be wtisfied only when

t .e . , when ( 8 x -.g- ) becomes parallel to 8.

Reflection of Electromagnetic Waves PZS

choosing a, new set of perpendiculitr axis, with 11, ill the direction of I;, we see L(S), p(S), y(S) are invitriantx for orthogone1 tmnsformations. Thr vector eqilntion can be split up into two equation8

So that remembering 82,+6,; = 0

we have

which breaks up into two equations

which can again be put in the form

total reflection is obtained as before by putting p = 0

as grd4 S can also become infinite, if 7 =

le?b f c m - V

i.e., for a frequenoy equal to the gyromagnetic frequency.

case 111.

Neither 4s ) nor 0°= 0

i.e., when \grad 86) f 0 and 8 ie not parallel to E.

We have to proceed to further elimination to get the refraotive index surfam.

The elimination work becomes simplified if we &oow a new a& of axis, with Z axis,'in the k t i o n a ~ f h.


The veator equation is now equivalent to the t h scalar equations

dS eh Ne* W V , - B ( S ) = - rn &(sA + sueu + SA) . . . ( i )

dS eh Ne* ~ ~ ) ~ , + q ( s , ~ ozo = - m S~(R,O.+S, ,~ , ,+S~~) ... ( i i )

Nea L(S)& = -;;; SASz~.+S&+SA) . . . (iii)

where ( a8 ds 8. ==, sv = --

a~

From ( i ) and ( i i ) we easily get the relations

... (iu)

... (v )

.. . (vi)

multiplying (io), (u) and (Vi) by S#, Sv, Sz, we have


Remembering Ne2

L(S) - -grad2S m = p(S)g(S)

and rejecting the factor g(S) the equation finally msumnes the form

dS When this equation is regarded as a quadratic in (grad2$) i.e., when the value of dt -

a s is supposed to be prescribed, the two roots of gradZS, become, let us say, f, (F , a )

a s and f2 {x, a 1. So that we get two rays corresponding to any prescribed a.

Corresponding to the two refraotive indices

a s When damping is neglected, - = L a. at

We see from (iv), (v) and (oi), 8, : 8, : 8, a s

defines a complex-ratio. As YS) will become real, and -pure imaginary, the two split at


rays will there be elliptically polarised, and t,he ratio of the axis, projected 011 a plane perpendicular of the magnetic field, is easily seen to be

eh for large values of p;rad2S this tends to - . cm y,1

Writing out the equation in extenso and putting grad2 S = y, we see the

equation

as as The co-efficient of y 2 = p(S) - as as as a, ( , + v ) ' + ( t ) " (, i [z ( -2r -4

The absolute term on the other hand is equal to

We get therefore

as defining the critical frequencies for total reflection. This is in t.he general wse when collision frequericy is not neglected an equation of the fourth degree having two different. conjugate complex roots corresponding to two elliptically polsrised rays: or when collision frequency 'is neglected,

the above relation redurns to

Reflection o f Electromagnetic Waves

which agrees with the expression arrived a t by Rai, Bajpai, and Mathur, and also with the result arrived a t by another method.

From energy consideration, we have seen that when E 11 0 the condition of total reflection grad S. [E x H) = 0, is satisfied (grad ~5')~ = 0 : Here the propagation stops from penetrating further, as H vanishes, and the field beyond becomes electrostatic

in character. When ( g a d S E) .f 0, which will happen when the direction of propagation makes an angle \vith the imgnetic field, the electric vector is not the wave front, and the total-reflection condition results when electric vector coincides with the wave normal. This happens as (grad S)2-m . The Poynting vector t,hen lies in the wave front. The flow of energy across the surface stops, and the waves get totally reflected.

CONCLUSIOS

We Iiave discussed the question of ionosphere a t some length, but i t will be easily hcen, that the n~ethod can be applied with the same ease for discussion of propagation of light in a material medium. We ha\-e only to utilise the corresponding Lorentz- equation for motion of bound-electrons, which can be made to yield a relation similar to the equation (Z) deduced for the ionosphere. Readers interested in mathematical analysis will easily recognise the method as an application of the method of characteristics, used for discussions of wave-propagation by Hadamard, Debye, and others. So far as the writer is aware the method has not been applied to the microscopic equations of Lorentz, where however the extension does not seem to create any difficulty when the waves are looked up011 as possible modes of oscillation of the stationary electro-magnetic field, generated by positive and negative charges, present in the medium.

REFERENCE

1. Bajpai, R. R. end Blathur, K. B., Znd. J . Phys.. I t , 165 (1937).

Reprinted from Ind J Phys 12, pp. 121-144, 1938 (Indian Association for the Cultivation of Science).

I. The decomposition of any Lorentz transformation iuto two comlnutable factors is a well-known result, and forms a convenient starting point for the thcory of spinors. Einstein arrived a t the decomposition from consideration of the. infinitesimal elements of the Lorentz group. The result is here obtained from purely algebraic consideration, which gives a t the same time a slightly more general result inasmuch as the factorisation can be shown to hold for elements of the general ort,hogonal group C4, with det C, = +l. The Lorentz group L is really a sub-group of C, inasmuch as the co-efficients are restricted by certain reality conditions, for every matrix C in L has C,, real (r, 8 = 1, 2, 3), while C4,, Cr4, r = 1, 2, 3 are pure imaginaries, and C, real.

If C is a matrix, the transposed matrix will be denoted by 8 ; a symmetric matrix

for which C = C will be denoted by 8. If C+C = 0, the matrix is anti-symmetric and will be denoted by A.

A non-symmetric matrix C = S + A while 8 = 8-A . We shall assume here the elements of a matrix to be complex numbers.

2. Prop. I. An anti-symmetric matrix A having six non-zero elements can be further decomposed into two component matrices P and Q, which have only three non-zero elements such that

If further we introduce certain basic matrices, defined by

Studies in Lorentz Group

in which the first set P have the properties of the above P matrices in ( I ) , and the second set Q those of the Q matrices. We note that

P ; = P ~ - ~ ~ - Q ~ - Q ; = Q ; = - E 2 - 3 - I -

P l = P 2 P 3 = - P a p 2 , P 2 = P 3 P l = - P I P 3 , P 3 = P l P 2 = - P 2 P l

Q1 = &2&3 = -Q3Q2, Q2 = &3&1 =-Q1Q3, Q3 = Q1Q2 =-Q2Q1,

pr Qs = QsPr. . .. ( 3 )

Any P matrix can t,hen be represented as

P = a,P,+a2Pz+a3P,, . . . ( 4 ) and any Q matrix RR

Q =-; hlQ, + b2Q2 +b3Q3. . . . (5) Hence

A = alp, +a,P2+a3P~+biQi+b2Qz+b,Q, ... (6)

Carrying out the multiplication of a11 Pi's and Qi's (i = 1 , 3, 3), we get in all six- teen matrices which we can utilise as basic matrices for the representation of any arbitrary matrix C thus (the indices of Q will be raised in what follows when Q's appear with summations)

Any symmetric matrix

Since Pf = - E, we have generally


Prop. 11. If S is any orthogonal matrix, i.e., S2 = E, det S # 0, then, either

( i) det S = -1, spur S # 0, or (ii) det S = +I , spur S = 0.

We take a symmetric matrix

S = l a h g 1:

I& b f Y

s f c Z

2 Y z d

Since S2 = E, S = 8-1, and as det S # 0, the inverse can be written down by utilising the minors of the det S from which the following results follow easily.

If s = det S , and A , B , C , etc. are minors of the determinant

D = l a h g

b f I : f c

then s(ad-x2) = A s( fd-yz) = F

~ ( b d -y2) = B s(gd -zx) = G

s(cd-22) = C s(hd-xy) = H

BC-F2 = an etc. etc.

9 , 1.

We also remember the usual orthogonality relations of the type

ax+hy+gz+xd = 0, ctc.

n2+h2+g2+x2 = 1 , C ~ C .

So we easily deduce Az+Hy+Gz+sx = 0

FZx+By+Fz+sy = 0

Gx+Fy+Cz+sz = 0,

so that, if x, y, z be not all zero, we have by elimination

s3+s2(A+B+C)+sD(a+b+c)+D2 = 0,

or, ns s2 f 0, and D = sd (s+A+B+C) = -d(a+b+c+d).

Again A - t B + C = s[d(a+b + c)-z2-?j2-z2]

= s[d(a+b+c+d)-11;

so that (A+B+C+s) = sd(a+b+c f d ) .


Hence either s = -1, or cc+b+c+d = 0 , or d = 0.

The special cases d = O and x = y = ' z = O

can be treated in the same way, and we will have in every case s = - 1 or spur S = 0, which proves our general result.

Prop. 111. If S2 = E and spur S = 0 (so that s = + I ) , then S is always de- composable into two commutative anti-symmetric factors.

As spur S = 0, we can write by (7 )

s = C s ; P h Q ~ Now putting

d w r e A,, are certain matrices of the P class, we have

we must have h2A3 = A3A2

A3Al = AlA3

A,A, = A,Al.

Hence the A matrices are comm~~table .

I t follows easily, as the A's ate all of the same P claw,

A, : A, : A, - k , : k, : k ,

where k,, k,, k , are certain number%:

hence

which puts the symiiletric orthogonnl matrix with zero spur into two commutative miti-symmetric fiictors.

Also

det = I det (11P,+12P2+13P3) I I deb (klQl+k2Q2+k3Q3) I


whence 1 = ( l ~ + l ~ + ~ ~ ) ~ ( ( k + k:+kE)2. ... ( 1 4 )

we oan put

Then

When the elements of S are given, we can determine the elements of the factor matrices in the following way.

As

hence

Also

so

, and n+b+c+d = 0 ,

S =

= klll + k212 + k3z3

n = kill- k21,- k31,, etc.

h = kll,+k24, z = k2El-kl12.

a h, g x

h h f Y

9 . f C Z

2 Y Z d


Hence,

whera also

3. Decomposition of Non-symmetric Matricm

A general orthogonal matrix can be written as

C = S+A

C G = C C = E . where

From this follows SA-AS = 0, 8'-A'= E.

where P and Q are linear combinations of P and Q type matrices,

we have s ~ - P ~ - Q ~ - ~ P Q = 8

L.c . , S" (1 +P2+Q2)E+2P&. Now

spur C = spur S = p (say), hence by (7),

S = pE+APQ. Again as

S2 = p2 + haP2Q2+ 2 / .dPQ.

we haw comparing ('20) atid ( Z l ) , firstly

Thus


Identifying (20) with ( 2 1 ) we put

which relation is equivalent to , u ~ + ~ ~ = I + P ~ + Q ~ . 16'

The numbers kl and k, are thus determined. Thus (23) which is the factorisation aimed at is completely established.

The general form of C cuLn also be calculated.

where

and

Explicitly,

A = A. -Aa A2 -Al

A3 A. -A1 --A2

-A2 Al A" -A3

A1 A2 4 A"


and spur C = 4AoBo. . . . (27)

Two special forms of C can be at once noticed :

If C is a metric of the proper Lorentz group, the11

C14, C,,, C,,, Cdl, C,,, C,, are pure imaginaries, so that

(A2B3 - A3B2) + (A1Bo - BIAo) = %I.

(A2B3-A,B,)-(AIBo-BIAo) = ih',

from which fo11ows

(A2B3- A3B2) as well as (AIBo- BIAo),

etc, are all pure imaginaries.

AoBo, A;Bl, A,B,, A,B3 are real quantities as well as (A,B3+A,B2), etc. and (A3Bo+AoB3), etc. are real.

We have then

A0 = A'_ - - - - A' = 2% = k, red ... B ~ * B,* B * B3*

.s A!+Af+Ag+A; = B,2+Bf+B$+Bg = $1 (Bo* representing the conjugate of B,, etc.).

226 S N Bose : The Ma n and His Work

We thus see that the sets (A,, A,, A,, A,) and (B,, B,, B,, B,) are conjugate to each other.

There are now two cases possible.

Case I. When (A,, A,, A,, A,) are real, then A. = Bo, Al = Bl, Aa = Bp, As = B3

and from (26) further C14 =L CM = cac = C4, = C4* = Ca = 0

c4, = 1.

This represents a three dimensional rotation.

The axis of rotation is easily seen to be

The angle of rotation is given by

Case 11. A,, A,, A, are pure imaginaries., A, real.

The resultant transformation is here seen to be an Einstein transformation,with translation along the line

and velocity given by

where A;--Ge = 1, and @ = Aq+A;+Ag.

4. We shell now consider certain general matrices of the form

and

We can easily see that

I det A I = @s+d+bs+c7s

I det B I = w+as+bs+cs]a.

So when a, b, c are oomplex quantities and I det A 1 and I det B I real and positive we must have


,u2+a2+b2+c2 = p*2+a*2+b*2+c*2

where p*, etc. are conjugates of p, etc.

We now consider A and B to be factors of decomposition of C, and take the determinant of each of the factors to be + l , so that

,u2+a2+b2+c2 = &l .

The matrices of A and B types are orthogonal matrices; but since their elements are generally complex, they do not belong to the Lorentz sub-group of Group C,.

We will now discuss certain general characteristics of the A and B matrices, which will bring into light also the intimate relation that exists between the decompositions as carried out here and the spinor theory.

Prop. All matrices of A and B types are reducible :

where 8 and Q are certain matrices of the second order. That such matrices are reducible is a well-known result.

define a transformation of a four-component + to a four component $.

We have

$1 = PO+l - ~ 3 $ 2 +p2+3-pi44

$2 = P3+1 +p0$2- ~ 1 + 3 - ~ 2 + 4

+3 = - ~ 2 + 1 + ~ 1 + 2 + ~ 0 ~ 3 - ~ 3 + 4

$4 = + ~ 2 6 z f pa#$ PO+^-

From this it can be easily deduced that

and there is alsu the corresponding conjugate relation obtained by putting -i for i , (33) can also be written as


Combining (33) and (34) we obtain

$3fi$4 $ l + V 2 P O + ~ P ~ ~ 2 - j ~ l

;+a- +I +3-i$4 l = l - P ~ - ~ P I PO- which is a well-known spinor transformation.

Again, choosing two sets of variables as (x,, x,, x,, x,) and (yl, y2, y3. y4) the corresponding transformation

y,+i?l, --?/l+i?l,

~1+i?/ , ?/3-i~4

formula for B matrices can be written as

which can be also written in the form

(35) can also be written in this notation with the transformat,ion matrix to the left on the right-hand side.

Thus A can be regarded as inducing a front transformation while B induces a back transformation, or vice-versa, which shows the commutative nature of the transformation a t once.

The general reducibility of the A and B types of matrices shows that the Lorentz Group can be studied advantageously in the two-dimensional representation of the A and B class of transformation.


THE COMPLETE SOLUTION OF THE EQUATION:

By S. N. Boss, Lkma University, and S. C. KAR, Bangabmi College, Calcutta.

(Received November 17, 1940.)

1. In what follows we desire to present what appears to us to be a complete Kirchhoff-like solution of the equation above and indeed by two distinct methods. The one is a method of complex integration and the other an extension of a method adopted by. Love * for ordinary retarded potentials. The equation itself is one which seems t,o have acquired some importance in view of a recent work of J. H. Bhabha t on the nlesotron, who has stated a solution of the equation, in which the part involving a s u r f ce integral does not appear. To the best of our knowledge and belief the solution at which we arrive and which exhibits the surface integral is original. As the whole work, however, was started by a successful derivation, by an adaptation of the well-known method of Herglotz and Sommerfeld $, of the potentials of a moving mesotron, such as Bhabha uses, we let this derivation precede the main investigation, since this way of arriving at the potentials appears to us to be also new and of sufEcient interest.

2. To st& with, we seek a solution of the equrttion

of the form: # = #(a),

where 82 = ~2(e-t)2-11([-~)2 = cq8-t)2-re . . . . (2)

and x, y, z, t are parameters. The transformation of eq. (1) to the single vsriable a leads to the equation

Henoe we infer: $ = J1(kr) - or - ''(w, where ~ ~ ( b ) and Y~(# are ~eaae l 8 8

funotiom of the first end second kinds respectively.

VOL. VII-NO. 1.

5

[Publiehed April Bth, 1941.


3. Retar& potentials of a moving mtsotron.

and c h o ~ e for the four potentials (4, Al, A,, AS) of the meeotron solutions of eq. (l), steted as an equation in x, y, z, t, with I , 9, f , 8 as parameters, in the forms

The equation of restraint on the four potentials becoxnee

if, after Herglotz and Sommerfeld, we treat I, 9, f as functions of 8 and put

. . V4,fi,fe,fs)=B a,a , . . (7) where B ie an absolute constant.

We assume now 8 to have been complex of the form p+iv and proceed to perform the integrations in eqs. (6) along a path on the complex plane shown in the diagram below.

i Y

The Complete Solution of the Equation :

The following considerations govern the choice of from eq. (2),

82 = Cz(o-t1)(6-t'), . . r r

where t' = t- - and t' - - t+; , the only singularities of x C

and 8 = tm and since we are seeking, for the moment, tho

95

this path. Since,

. . . . (8)

are tho~e at 8 = t'

retarded potentials - we let the path embrllce just the one singularity at 8 = t'. Since, further, Ins in x is multiple-valued we cannot satisfy the equation of constraint (6) by just having a closed contour about t' but have to let the path run along the real axis from -a, on one Riemann surface round t' to -a on a second, as, with 8-t -a, x -+ 0.

Retaining only such terms in x as involve a singularity we may write eqs. (5) aa

etc., and evaluate the integrals as follows:

1 df !hi d ~ ' ni v,, eto.,

-00

We identify now Bni with the charge e of the mesotron and obtain as its potentials

whioh are the forme ueed by Bhabha.


In concluding this section we remark that the advanced potentials of the mesot,ron, also used by Bhabha, may be o b t h e d in like manner by the choice of a path of' integration made to run along the real axis from +a on one Riemam surfwe round the other singularity at 8 = t' back to +a on a second.

4 . Solutimt of the proposd equation for rehrded mlwa of t : First method. We write

9 4 v29 - -k24 = -4np(hV) . . .. (10)

iind Vex - &ij@ px -kS( =O,

where x is given by (4). Hence we obtain firstly,

and then, upon in tqpt icn with reapect to 8,

We assume now 0 to have been complex of the form p+iv and proceed to perform the integrations along the same path on the complex plane es we have used in the previous section. Since, with 8-+ -a, both x and

% -r 0, we have ae

This we may write again in the form

if we observe that we may treat the path of integration es independent of f ,q , 1, since the path about the eingularity at t' may alwaya be slightly varied

a to acaommodate the shift of t' involved in the operatione, - etc.

We eet down now in brief the neoeseery cslculations. ats


I:: (iii) d o + % = - ( I - % )


m' Substituting these results in eq. (11) and dropping the factor - we get

C

We integrate this equation now further through a closed volume about the point P (2, y, z) but exclude the volume of a small sphere about that point. We obtain thus

where the seoond surface integral is over the surface of the amdl spherci about P. I ts limiting value, with r -t 0, is seen to be


We arrive henoe a t the oomplete Kirchhd-like solution for retarded values of t in the form

I n closing this section we remark that a like procedure with the second path of integration pointed out in the previous section yields the complete Kirchhoff-like solution for advanced values o f t in the form

5. 8ohtion of the propo8ed equation for retard& vah?.s o f t : 8econd method. r

To begin with, we replace 8 by t- -- (= t') in eq. (10). With the notation C

etc.,

equation (10) take0 then the form

S N Base : The Man and His Work

Now

Hence we deduce firstly

and then

= -4+] . . . . . . (18)

in view of eq. ,(16). Dividing both sidee of eq. (18) by r, we have, 1

sin06 VS- =0, r

a l a a I 2 1 a r a kY+I bl - -&r-. . . ( A ) c%{; q[+~-[+~&) + ; ?- c

We nexfi proceed to oombine eq. (10) with

w h Q=- Jl(k-9) 8 '

and obtain firstly

and then, .inw both Q and 3+0 with e+ -my ae

The Complete SolutMn of the Equation :

Rearranging the terms in this equat'ion we write

J -m

and observe that

k [Ql = ij ,

and, in view of eqs. (17),

Equation (19) now takes the form

Combining now equations (A) and (33) we get


or, in view of eqs. (IT), 1'

a 1 a+ 1 ar a+ 2,[;[s~+;~[;.]-[41;(a-d/ - ID ae(~p-+g)

This equation (20) is easily seen to be identical with eq. (12) and snbsequent integration through a closed volume, as in t,he last section, will now obviously lead to the complete solution of the proposed equat,ion in the form (13).

I t is evident that we niily in like manner obt,ain the complete solution T

for advanced values of t if we replace4 in eq. (10) fib, the outset by t +- ( = t " ) C

and later perform integrat.ions with respect to B from t" to + m. 6. In closing this communication we'like t'o make a fern observations

on the methods here employed. Firstly, the method of complex integration is obviously also available for the Kirchhoff solution of the equation

We may have this solution, of course, immediately out of the solutions (13) and (14) with k -t 0; we may have it, t,oo, by a like procedure repeated with a

1 solving function ~ ( s ) = - Secondly, both methods may be adapted without

82 '

difficulty for solution of the equation

and the only difference should be that real Bessel functions of imaginary arguments would take the place of those we have used above. Lastly, underlying. the method of complex integration there is obviously the assumption that p and 4 are analytic functions of B over a region of the complex plane wide enough to embrace the path of integration-an assumption from which the second method is happily free.

* Love: Mathematical Theory of Elasticity, 4th Ed. (1927). pp. 302-3 or Lond. Mnlk. Soc. Proc., Ser 2, Vol. 1, (1W4).

J. H. Bhabha: Proc. Roy. Soc., Vol. 172, p. 384, (1939). Herglotz and Sommerfeld: Frenkel'a Lehrbuch tler Elebtrod~/nnn~ik, Bd. 1 , p. 177,

(1920). 5 Forsyth: A Treatise on Diffewntial Equatioila, 2nd Ed., p. 167.

Reprinted from Proc Nut lnst Sc India 7, pp. 93-102, 1941 (Indian National Science Academy).

REACTION OF SULPHONAZIDES WITH PYRIDINE : SALTS AND DERIVATIVES OF PY RIDINE-IMINE

Curtius and Kraemersr heated p-toluene- sulphonazide with pyridine and isolated a crystalline compound, M.P. 2 10°C. They supposed it to be p-toluenesulphon-amido-pyridine. When, however, the sulphonic acid residue was removed by hydrolysis, the new base yielded a picrate of M.P. 138-9°C. which did not agree with M.P. of the picrate of any of the known amino-pyridines. The structure of this, compound was thus not established by this work of Curtius.

Reactions of p-acetyl-amino-benzene- sulphonazide and p-toluene-sulphonazide with pyridine have been studied here for the last year and a half. This has led to the discovery of a new derivative of pyridine, N-imino-pyridine, and the compound of Curtius has now been shown to be a sulphonic acid derivative of this N-amino-pyridine or pyridine-imine.

(1) When p-acetyl-amino-benzene-sulphonazide is

boiled in dry pyridine in an inert atmosphere, nitrogen is slowly evolved and from among other products a crystalline compound insoluble in pyridine can be isolated. After repeated crystallisation the substance melts with decomposition at 283"-4'C. The analysis shows that the substance has the same empirical formula as p-acetyl-amino-benzene- sulphonamido-pyridine but closer investigation reveals it to be also a derivative of pyridine-imine having the formula

(11) The acetyl-group can be removed easily and this

process leads to an extremely hygroscopic hydrochloride of a base. The free base has M.P. 228- 229°C. Estimation of Pt. in the platini-chloride compound, M.P. 224'C, indicats the formula

(CI lHl102N3S)H2PtC16 showing that it has two basic groups.

When further hydrolysed with hydrochloric acid, sulphanilic acid separates out and the solution of an

extremely hygroscopic hydrochloride of a base is obtained.

The perchlorate of this base has M.P. 204'C., the picrate, 149°C. and the platini-chloride, 237°C. and the analyses completely agree with the expected formulae.

Attempts to liberate the free-base with alkali lead to polymerisation. ' Alkaline-ferricyanide liberates nitrogen from the salts of this base. Nitrous acid decomposes the salts and from the solution pyridine can be isolated, as perchlorate.

The same base is obtained from the hydrolysis of paratoluene-sulphonazide-compound. This has been confirmed by allowing p-acetyl-arnino-benzene- sulpho-chloride to react on the base from (I) which leads to the compound (11).

A confirmation of the imino-structure has been obtained by a synthesis of the compound (I), from glutaconic-dialdehyde. Mono-benzoyl-derivative of glutaconic-dialdehyde2 is allowed to react with paratoluene sulpho-hydrazide leading to the compound of the form

CgHgC00 CH = CH - CH = CH - CH = N.NHS02C7H7

With the action of alcoholic hydrochloric acid the benzoyl group is eliminated, and there is a ring closure leading to the formation of the pyridine ring, and the hydrochloride of compound (I). This synthesis is similar to the synthesis of pyridine-oxide by Baumgarten.3

Further investigations about this interesting class of compound are in progress. Full details will be published elsewhere.

In April number of Current Science, Ganapathi and Miss Alamelas reported about their attempts to prepare sulphonamide derivatives of heterocyclic compounds by Curtius process. They mentioned the com und (11) and assumed this to be a 3-amino- pyri S" ine derivative without any justification. It is expected that this note may be of some interest to them.

S.N.BOSE PARITOSH KUMAR DVTTA

Dacca University, Dacca, 1-6- 1943.

1 Curtius and Kraemer, 3. Prakf. Chem., 125, 323, 1930. 2 Baumgarten, Ber., 57, 1625, 1924. 3 66, 1808, 1933. 4 Ganapathi, Current Science, 12. 119, 1943.

Reprinted from Sci & Cult 9, pp. 48-49, 1943 (Indian Science News Association, Calcutta).

A NOTE ON DIRAC EQUATIONS AND THE ZEEMAN EFFECT

BY S. N. BOSE AND

K . BASU (Receieed for pzdblication, Sept. 22, 1943)

ABSTRACT. A new treatemat has been given for solving Dirac's equations for hydrogenic atoms, and the radial functions are expressed in terms of a combination of SOPlbe's

polynomials T':) (p), TA!; (p ) of only two ~onsecutive degrees n, n i l ; and the elementary properties

of such polynomials have enabled us to tackle the Zeeman effect problem in general (homogeneous field) leading to the standard quadratic equation in energy for the effect.

1. With the help of the two-dimensional n~atrices s,, s,, B,, of Pauli the wave- equations of Dirac can be put in the well-known matrix form :

where D is the operator a a a 8 , - +8 - f St-

\ ax * ay a%,

I If, similarly S = m,+ys,+zs,, and s = -(zs ,+ys,+q,) , then T

d 8.D. = r zr+L, where L = 1 (Mg,+M$,+Mb,) .

also

And the following commutation rules can be easily deduced :

8(L- l )+ (L- -1 ) s = 0, D ( L - l ) + ( L - l ) D = 0 ;

a2 = 1.

Hence multiplying the equations on the left by 8 we have

Note on Dirac Equations and Zeeman Effect 24 1

O; Y = 121 = Y where O, are similar ons- r

column matrices and functions of (8, #), the equations can be rewritten as

The equations become easily separable if the matrices @ and Y are ;so chosen that at first (L-l)Y = kY and SO = Y; and as s2 5 1, it follows therefore from commutation rules that

@ = sY and (L-1)sY ---ksY.

We observe a t once that L(L- 1) Y = - 2 Y = k(k+ l)Y, where

1 a a 1 82 o - - - ( sin 0 a) + ----L . = in 8 do ~in2 8 a@. '

and therefore k can be either a positive or a negative integer.

Secondly, if (@, Y) are the matrices for positive k, then (Y,O) are the matrices for negative k. 'Also remembering that the operator

a 8 1 - [ L ( z - - - ) - ] comnutes with the equation-system, the angular

dy dx

matrices can be expressed in terms of spherical harmonics of order k and k- 1 in the following form :

k-m ,/- Y$ 2k+ 1

(k, a positive integer) (1.4)


The functions are interchanged for negative values of k. \Vriting f ( r ) = iF(r)

so as to remove the imaginary from the radial equations, we get

2. To solve the equation we assume

F = foe-"rPFl(r), G = goe-ArrWl(r), and N = aE/(Eg-E2)i . .. (2.1) where Ze2/hc = a.

Substituting in (1.5), F, and G, are easily seen to satisfy the following equations :

Ahc where fo/go = -(Eo-E)/hcA = ---- = E E t Ea-E2 4

EoSE - ( E E a n d = ( - ) (1.3)

If Fl+Bl = 2X+, (2,-F, = 2 ~ - , elimination leach easily to the following equa- tiona :

provided kS-pa = Na(q-E2)/Ea = a2.

Taking the first equation it oan be easily seen that it admits of polynomial solu- tiom if N-,u = n (an integer). Writing p for Shr, and writing a S o h e polynomial in the form

Note on Dirac Equations and Zeeman Effect

with constant Bc:) so chosen that it is normalized according to

we get B,") = 1 /{lt ! I'(n+2p+ 1 )))

Hence it easily follows that ,y+ and X- will have solutions as given below :

where the normalising hctor 6 is to be determined presently.

Our d i a l f~uwtions of eqs. (2.1) stand thus

Normalisation requires

which gives

We can write tho two solutions (A) and (B), corresponding to positive and negative values of k respectively.


wherein it is understood that f- and g- are obtained by changing the sign of k, in the expressions for F and in (1.5) above.

3. When the atom is in an electromagnetic field defined by the vector potential ( A A f i , ) , the Dirac-equatiom become

In the case of a constant magnetio field H, in the direction of .Z-axis

We observe in passing that in the absence of A, the wave function X is generally small compared with Y-the radial component of F(r) has the factor

E e J l--' , while G(r) has ,,/ I + - ; so that the perturbation effect of - AX is Eo Eo hc

smell as compared with effect duo to the term -% AY in eq. (3.1) h.c

we seek an approximate solution of the equation by choosing one set of angular func-

Note on Dirac Equations and Zeeman Effect 245

tion (Y;, Y;+l), and assuming the existence of both the sets (YL, , Y;!,') and (Yk, , YgJ) in X. This can be done by suitably combining the (A) axid (B) types, thus

Take uj = Cf213++C=(1+~) v;, (j = 1, 2, 3, 4), where (u;, u;, u:, uf) corresponds to k = +I and (v?, v;, v;, v;) %orrespends to k = -(1+ 1). More explicitly, their values are

The constants A, N, p of the two types of solutions are different and are expressed by the following relations

2 A+ = (Et-P+)+/hc, la-ua = p,, N+ = aE+/[El-E?]+, (3.4)

and N+ =n++p+[for (A)-type with k = 11.

and N- = n-+p-[for (B)-type with k = -(1+1)].

If E+ and E- differ slightly from one another, we have the following approximate relations

By following the usual method of perturbation we see easily that the characteristic equation for determining the Eigen value E would be

where p = eH/2hc, and


a,, = [ H t r a Y++ Y ; m X + ] d V

a,, = [ H l r a Y - + Y l r a X - I d V

and X; etc. are transposed conjugates of X+ (the a is given in (3.3))

Carrying out integrations over the angle-variable-space we get

(ID

a, = 2 ~ ( 4 ~ + 2 ) ( J + 1 ) I r f - ~ - d r ; (%l+ 1 )W+ 3)

4. To evaluate the integral CI) or (II), we substitute the valuea of the c ~ m ' ~ o n d - ing f , g, and remembering r = p/2h, we obtain

(ID

A+ E+ (%-@+I+ {I)=-- 3 -E ) 1 e - p e + l [ f i# ) (p ) ]hp 2N+ 4ha { ( E+ 0

The integrals are evaluated quickly by repeated partial integration; thus

=(n+l)(n+2p+l)--n(n+2p) = 2N++ 1.

Similarly

Note on Dirac Equations and Zeeman Effect

(ID

[ 1 2 J e-pp%+l T$ll (p ) dp - 2N+- 1.

0 I -

An easy substitution of tho value of A+ in the foregoing loads to the following results :

The evaluation of the third integral cannot, however, be exactly expressed in a neat form ; for the simple reason that A+, N+, E+ as weH as A_, N-, E- are different in the twosolutions, as also p+ = 2A+r and p- = 2A-r. If however, we introduce approximations in the beginning, and take

{I111 = - 2 1 A l l E l ( E ~ - I E 1 2 ) t N%-l E0 i + l )'x 2 N 4111' ( B ) ' ( N z +

(ID

t e - p p 2 * + z ~ p ( p ) ~ k + (p)ap-- ( N ~ + z ) ( 4 - i - l )'

0

The two definite integrals can be easily evaluated in the same way as before. give the results below ;

Q (ID

e-~pn+2fi

J e-ppw2 T?) (p ) TWI) (p)dp = I (4.4) o o d n - l ! n ! F(n+2p+1) l?(n+2,u+2)

$ N Bose : The Man and His Work

.--A 1 h,

{III) = - - --- [ 4sS - r J.X%+Z+I \/~x/ , , / ~ + l + , ] SN Is',, /.: E

Observing E,/E - 1, we scc

Finally. putting 1) = cH!?lr,. and ek/2mnc = pn (Rohr-magneton) nntl subst,ituting the values of the three inliypds (and making E,,/E-I) we scc that tlre drterminantal cyantion (3.7) takes the form :

wllcrein we have put / t+d = m (magnetio quantum number).

The result (4.6) agrees completely with that quoted by Bethe (1033), supposed to have bccn worked out by llim from Pauli's equations. Condon and Shortley (!93;3) obtnined siniilar determinant from principles of quanturii mechanics by applying twofold pcrturbations (spin-orbit and magnetic) simultaneously. It may bc noted that Darwin (1928) has many similar features with our mode of attack, and wc cnn claim Rome elegance by our introduction of Sonines properties, which exhibit our solutions in good perspective.

R a r ~ a l c a c r e e

Bathe. H., 1933, Handbuch der Phydk, 2 1 p 396.

Condon, E. U. and Shortley. a. H., 1935. T h Theory of Afomic Spctm (Cambridge University Press), p. 233.

Dmwin Q. C., 1928, Proc. Roy. Soc., London (A), 118. p. 664-680.

Reprinted from Ind J Phys 17, pp. 301-308, 1943 (Indian Association for the Cultivation of Science).

Thirtyfirst Indian Science Congress DELHI, 1944

PRESIDENTIAL ADDRESS CON~RESS PRESIDENT : PROFESSOR S. N. BOSR

THE CLASSICAL DEX"I'RM1RTISM AKU'D THE QUANTUM THEORY (Delivered or8 Jan. 3, 1944)

I wish to express sincere thanks for the great honour you have done me. The Presidentship of tlie Science Congress is a great distinction, and I confess, I have my own misgivings about the wisdom of your choice. Your first decision had raised high hopes. Many of us expected that a deliberate p r o g r a n ~ n ~ of the future scientific activities of the country would probably be a feature of the opening speech of this Congress. Pandit Jawaharlal liad studied tlie needs of the country. Many of our front-rank scientists and industrialists liad met under his leadership, not long ago, and given to questions of future reconstruction much time and anxious thought. The results of this deliberation would have been invaluable at the present moment. My regret is keen that chance has deprived us of the benefits of a sustained and careful study -of the problems of the day. I would have liked to present here the results, if they were available. Unfortunately they are not, as most of the reports are inaccessible to me.

One of your foriner Presicleiits had reinarked that "a scientist is apt to become a nian that kno\\-s more and more about less and less, so that his opinion up011 subjects outside his field of special study is not necessarily of special value". I realise the wisdom of this warning and hope to have your indulgence, if I seem to he more at home with doubts and criticisms than with useful knowledge.

I would likc to present before you certain aspects of modern physics and draw your attention to the profound changes in the principle of scientific explanation of natural phenomena brought about by the quantum theory. The last fifty years record reinarkable discoveries. I need only mention the electron and the neutron, X-rays and Radio-activity to remind you of the increase of our knowledge. Our equipment has gained in power rangc and accuracy. We possess powerful telescopes to scan the furthest corners of the universe, also precise and delicate instruments to probe into the interior of the atoms and molecules. The alchemists' dream of transmutation has become a reality. Aton~s are now dis- integrated and synthesisecl. X-ray reveals invisible worlds and wireless links up the furthest ends of the earth with possibility of immediate inter- communication. These discoveries have their repercussions in the realm of ideas. Fifty pears ago the belief in causality and determination was absolute. To-day physicists have gained knowledge but lost their faith. T o understancl properly the significance of such a profound change it will be necessary to discuss brieflv how it all came about. Classical physics had begun with the stndy of astronoii~y. \frith his laws of gravitation and his dynaniics Ke\\:ton had explained planetary motion. Subsequent study has slio~\-n astronomical prediction to be possible and sure. Physicists had taken the equations of celestial mechanics as their model of a universal Ian-. The atomic theory had in the mean time gained


u~liversal acceptance ; since matter had resolved into a conglomeration of particlcs, the ideal scheme was to explain all phenomena in terms of their motions and interactions. I t was only necessary to set up a proper set of equations, and to take account of all possible mutual interactions. If the mass, position, and velocity of all. the particles were known at any instant, these equations would theoretically enable the physicist to predict the position and motion of every particle at any other subsequent moment.

The phenomena of light did not at first fit into this simple scheme. To regard it as a stream of particles was impossible due to the discovery of interference. Accordingly the wave theory of light was originated by Huyghens and perfected by Naxwell. With the discovery of the electron as a universal constituent of matter,'the electronlagnetic theory of Maxwell was converted into an electronic theory by 1,orentz. To the dynamical laws were added the electromagnetic equations and the two together apparently gave an exact and ideal formulation of the laws of causality. In the forces of interaction henceforth, were to be included not only the gravitatiollal forces but also those interactions which depended on thc charge and the nlction of the particles. These interactions were brought about by influences which spread out as waves with the velocity of light. They superilnposcd, interfered and constituted thc field of force in thc neighbourl~ood of the l)articles, niodified their motion and were in turn modified by them. Thc motions of all particles throughout the universe were thus interlocked. These out-going influenccs also constituted light, invisible radiation, X-rays and wireless ~vaves. Thus a set of universal laws was supl)oscd to have been discovered and we had only to apply them suitably to find explanations of all conceivable natural phenomena. In physical scictice wc do not however always proceed in the above way and turn to the 'rmicroscopic" equations whenever we have to explain events. We often study materials en mnsse, consisting of an enormous tlumber of corpusclcs, and we use either the principle of the conservation of energy or the l a w of thernlodynanlics to explain their behaviour. These laws \Yere however regarded either as simple consequences of the fundamental equations or as statistical laws derivable from them by a suitable averaging. Though in the latter cases we talk about probabilities and fluctuations, it was more or less a matter of faith to maintain that if it were possible for us to obtain all the necessary data by delicate observations, universal laws wourd enable us to follow each individual molecule in this intricate labyrinth and we should find in each case an exact fulfilment of the laws and agreement with observation. The above in brief forms an expression of faith of a classical physicist. We see that it involves as necessary consequences, belief in continuity, in the possibility of space-time description of all changes and in the existence of universal laws independent of observers which inexorably determine the course of future events and the fate of the material world for all times.

A few remarks about the general equations will perhaps enahle us to follow better the criticisms that have been levelled against the system. The structure of the mechanical equations of particles is different from the field-equations of Maxwell and Lorentz. The principles of conservation of energy and momentum were first discovered as consequences of the mechanical equations. Mass and velocity of the corpuscle furnish means to measure its momentum, and its energy, if we leave aside the potential energy which resides in the field. To maintain the integrity of the principle of conservation, the field must also b e considered capable of possessing energy and momentum, which however, being associated with wave-motion, must spread out in all directions with the waves. The transfer

Cla.ssical Determinism and Quantum Theory

of energy from the field to the particles must thus be a contiuuous'process, whereby, a finite change should come about only in a finite interval and the process should theoretically be capable of an exact description in space and time.

Physics being essentially concerned with relations between quantities, these should all be capable of exact measurement. We measure al~~rays intervals of time or inter-distance between points, hence the specification of the reference frame is just as important as the units of measure. Newton had not analysed closely the conception of mass and time. This vague- ness persisted in the dynamical equations for the particles. The field- equations which form the basis of the wave-theory of light have a different- origin. With the discovery of the principle of the least action, a con~mon derivation of both has been attempted. But a difference in the choice of reference frame in the two apparently subsisted. The wave-equations assumed a fixed ether whereas the material 1au.s conteliq)laiecl a Galilean inertial-frame. An immediate deduction from this distinction \\.as the possibility of measuring the relative velocity of the observer with rcfcrence to ether. The experiment of Michelson and Morley showed it to be unrealis- able in practice and formed the starting point. of the celebrated Relativity Theory. Einstein had subjected the conception of time-measurement to a searching examination and showed the impossibility of conceiving a timc independent of an observer, or an absolute simultaneity of events happening at two different places. The same space-time reference should be chosen for the dynamical equations as well as the equations of the field, this being supplied by the observer. In spite of this apparent limitation Einstein demonstrated the possibility of formulation of natural laws independent of all axes of reference and pointed out that the necessary auxiliaries existed already in the invariant theory and the tensor Calculus of mathematicians. In spite of its apparently revolutionary character, the theory of relativity upheld the ideal of causality and determinism. Einstein liinl- self has continued to seek with great earnestness a unifying field theory which will combine gravitation and electromagnetism and render unnecessary a separate formulation of the dynamical equations. No such theory as yet exists.

The developnient of the quantum theory has raised fundanmltal issues. Facts have been discovered which demonstrate the brealtclown of the fundamental equations which justified our belief in determinism. A critical examination of the way in which physical measurements are made has shown the impossibility of measuring accurately all the quantities necessary for a space-time description of the motion of the corpuscles.

Experiments reveal either the corpuscular or the wave nature for the photon or the electron according to the circumstances of the case, and present us with an apparently impossible task of fusing two contradictory characters into one sensible image. The only solution suggested has been a renunciation of space-time representation of atomic phenomena and with it our belief in causality and determinism.

Let me briefly recapitulate the facts. In rgoo Planck discovered the quantum of action while studying the conditions of equilibrium between matter and the radiation i3eld. Apparently interchange of energy took place in discrete units whose magnitude depended on 'h' and the frequency of the radiation emitted or absorbed by matter. Photo-electric emission had similar disquieting features. Einstein therefore suggested a discrete structure of the radiation field in which energy existed in quanta


illstead of beit~g continuously distributed in space as required by the wave- theory. This light-quantum however is not the old light-corpuscle of Newton. The rich experimental materials supporting the waveatheory preclude that possibility altogether. Moreover the fundamental relation, E=hv, and + = I & , connecting energy and momentum of the photon with the frequency v and the vector wave number k , makes a direct reference to idealised pla~ie wave so foreign to the old idea of a corpusclc. Soon afterwards Boiir postulated the existence of radiationless stationary states of atoms and showed how i t led to a sin~ple explanation of the aton~ic spectra. The extrenle simplicity of the 1)roposed structure and its striking success in correlating a multitude of experimental facts at once revealed the inadequacy of the ordinary laws of mechanics and electro-dynamics in explaining the remarkable stability of the atoms.

Thc new ideas found application in diffcrent branches of physics. Discontinuous quantum processes furnished solutions to many puzzles. Suitably modified, the theory furnished a reasonable explanation of the periodic classification of elements and thermal behaviour of substances at low temperature. Therc was however one striking feature. I t was apparently impossible to characterise the details of the actual transition processes from one stationary state to another, that is, to visualise it as a continuous sequence of changes determined by any law as yet undis- covered. I t became clear that the dyliamical laws as well as the laws of electroi~~agnetis~m failed to accouut for atomic processes. New laws had to be sought out compatible with the quantum theory capable at the same time of explaining the rich experimental materials of classical physics. Eohr and his pupils utilised for a time a correspondence principle, guessing correct laws for atomic processes from analogy with the results of the classical theory. In every case these appeared as statistical laws concerned n-it11 the probabilities of transition between the various atomic states. Einstein tackled the problem of the equilibrium of matter and radiation on the basis of certain hypothcses regarding the probabilities of transition between the various states by absorption and emission. A derivation of the Planck Law was obtained by Bose by a suitable modification of the methods of classical statistics. Heisenberg finally arrived at a satisfactory solution and discovered his ~natris-mechanics and a general method for all atomic problems. Dirac and Schrodinger also published sinlultaneously their independent solutions. Though clotlied in apparently dissimi1:r mathematical symbols, the three theories gave identical results and have now come to be looked upon as different for- malisms expressing the same statistical laws.

I have mentioned that the photon gave a simple esplanation of,,many of the properties of radiation and thereby presented its corpuscular Aspect wliile the well-known properties of interference and superposibility brought out its wave character. That the same dual nature may exist in all material corpuscles was first imagined by De Broglie. His phase-waves found quick experimental verification, and raised a similar problem of the real nature of the corpuscle. The formulation of wave-mechanics by Schrodinger, once raised a hope that by a radical modification of our usual ideas about the corpuscle it might be possible to re-establish the law of causality and classical determinism. Subsequent developments have shown such hopes to be illusory. His waves are mathematical fictions utilising the multidimensional. representation of a phase-space and are just as incapable of explaining the individuality of the electron, as the photon is incapable of explaining the superposibility of the field. The true meaning of his equations appears in their statistical interpretation.

Clnssicnl Determinism and Quantum Theory

111 The adherents of the quantum theory interpret the equations in a

peculiar way. They maintain that these equations make statements about the behaviour of a simple atom and nothing more than a calculatior~ of the probabilities of transition between its different states is ever possible. There is nothing incomprehensible about such a statistical law even if it relates to the behaviour of a single particle. But a follower of deter- ininism will interpret sucli statements as hetraying imperfect knowledge, either of the attendant circumstances or of the elementary laws. We may record the th row when a certain die is cast a large number of times and arrive at a statistical law which will tell us how many times out of a thousand it will fall 01; .L certain side. But if we can take into account the exact location of its centre of gravit!., all the circuinstances of the throw, the initial velocity, the resistan1.e of the table and the air and every other peculiarity that may affect it, therc can be no question of chance, because each time we can reckon wherc tlie die will stop and know iin what position it will rest. I t is the assertion of the impossibility of even conceiving such elementary determining laws for the atomic system that is disconcerting to the classical physicist.

Von Neumann has analysed the statistical interpretation of the quantutn mechani~al laws and claims to have demonstrated that the results of the quantum theory cannot be regarded as obtainable from exact causal laws by a process of averaging. He asserts definitely that a causal esplanation of quantum mechanics is not possible without an essential modification or sacrifice of some parts of the existing theory.

Bohr has recently analysed the situation and asserted that we c:ii~not hope any future development of the theory will ever allow a return to a description of tlie atomic phenomena more conformable to the ideal of causality. He points out the importance of the searching analysis of the theory of observation made by Heisenberg, whereby he has arrived at his famous principle of indeterminacy. -4ccording to it, it is never possible for us to determine the siinultaneous values of momentum, and positional co-ordinates of any system with an accuracy greater than hat is com-

h patible with the glequality A> nq> G.

This natural limitation does not affect the physics of bodies of finite size but makes space-time descriptions of corpuscles and pl~otons impossible. When we proceed to study the behaviour of the elementary particles, our instruments of measurement have an essential influence on the final results. We have also to concede that the contributions of the instrument and the object, are not separately computable from the results as they are interpreted in a classical way with the usual ideas of co-ordinate and momentuln accepting thereby a lack of control of all action and reaction of object and instrument due to quantum effects.

I t is in this imperative necessity of describing. all our knowledge with the usual classical ideas, that Bohr seeks an explanation of the apparently irreconcilable behaviour of corpuscles and radiation in different experj- ments. For example, if we set our experiments in sucli a fashion as to determine accurately the space-time co-ordinates, the same arrangement cannot be simultaneously used to calculate the energy momentum relations accurately ; when our arrangements have pushed the accuracy of determining the positional co-ordinates to its utmost limit, the results evidently will be capable only of a corpuscular representation. If, on the other hand, our aim is to determine momentum and energy with the utmost accuracy, the necessary apparatus will not allow us any deter-


niination of positional co-ordinates and the results we obtain can be understood only in terms of the imagery of wave-motion. The apparently contradictory nature of our conclusions is to be explained by the fact, that every measurement, has an individual character of its own. The quantum theory does not allow us to separate rigorously the contribution of the object and the instrmnent and as such the sum total of our knowledge gained in individual cases cannot be synthesised to give a consistent picture of the object of our study which enables us to predict with certainty its behaviour in any particular situation. We are thus doomed to have only statistical laws for these elementary particles and any further development is not likely to affect these general conclusions.

I t is clear that a complete acceptance of all the above conclusions would mean a complete break with the ancient accepted principles of scientific explanation. Causality and the universal laws are to be thrown simultaneously overboard. These assertions are so revolutionary that, no wonder, they have forced physicists to opposing camps. There are some who look'upon causality as an indispensable postulate for all scientific activities. The inability to apply it consistently because of the limitations of the present state of human knowledge would not justify a total denial of its existence. Granted that physics has outgrown the stage of a mechanistic formulation of the principle, they assert that it is now the task of the scientists to seek for a better formulation. Others of the opposing camp look upon old determinism as an inhuman conception, not only because it sets up an impossible ideal, but also as it forces man to a fatalistic attitude which regards humanity as inanimate automata in the hands of an iron law of causation. For them the new theory has humanised physics. The quantum statistical conception of determinism nestles closer to reality and substitutes a graspable truth for an inaccessible ideal. The theory has brought hope and inspired activity. I t constitutes a tremendous step towards the understanding of nature. The features of the present theory may not all be familiar but use will remove the initial prejudice. We are not to impose our reason and philosophy on nature. Our philosophy and our logic evolve and adjust themselves more and more to reality.

In spite of the striking successes of the new theory, its provisional character is often frankly admitted. The field theory is as yet in an unsatisfactory state. In spite of strong optimism, difficulties do not gradually dissolve and disappear. They are relegated to a lumber room, whencc the menace of an ultimate divergence of all solutions neutralises much of the convincing force of imposing mathematical symbols. Nor is the problem of matter and radiation solved by the theory of complementary characters. Also we hear already of the limitations of the new theory encountered in its application to nuclear problems.

The quantum theory is frankly utilitarian in its outlook ; but is the ideal of a universal theory completely overthrown by the penetrating criticism of the nature of physical measurements?

Bohr has stressed the unique character of all physical n~easurenients. We try to synthesise their results and we get probabilities to reckon with

h a+ instead of certainties. Bnt how does the formalism - -=h$ emerge 27ri at

as a certain law? The wider the generalisation, the less becomes the content. A universal law would be totally devoid of it. I t may nevertheless unfold unsuspected harmonies in the realm of concept. More than ever now, physics does need such a generalisation to bring order in its donlain of ideas.

Reprinted from lJruc lrrd Sci (hng 31, pp.1-6, 1944 (Asiatic Society of Ijcngal)

ON AN INTEGRAL EQUATION ASSOCIATED WITH THE EQUATION FOR HYDROGEN ATOM

BY S. N. BOSE

(Received on May 16, 1946) I

The SohrXnger funotione '#' oharaoterie'i the stationary atatea of hydrogen atom are now very familiar things in analysis aa aleo the differential equation whioh they aatisfy, namely,,

The aseooiated funotions M, defined by the relation,

oan be utilised for defining the probability in momentum spaoe if the momentum variables are introduoed by the relation p, = My p,, = hm, pz = hn.

When invemion ie poaeible, (1,2) implies also

and solutions of (1,l) can be utilised to oaloulate M's. Eleaeeer (1933) has followed thie method and arrived at fairly oomplioated formula. Ahother alternative would be to set up an appropriate equation for them, and invegtigate ite solutions. This ie an integral equation, whose oomplete mlution ie presented here. Tbs analpis presente several i n t ee ing features, and leads to expressions of M's, whioh o m be immediately utiliaed to study their properties or to apply them to physioal problem.

If we use the semisonvergent in-

then oornbination with (1,2) leads to the following sex-tuple integral for +/R, after a ohange of variable and order of integration :

M(A-1, p-m, v-n) exp. 2ni(Ax+py+ vz)dl dm dn dh dp dv.

256

M and


can thus be regarded as Fourier transform of q5 and #/R.

Using then in the Schrijdinger equation ( 1 , l ) we deduce the following integral equation

M(1-x, m-y, n-z) (la+ma+n2-k2)M = A dx dy dz, . . . (A)

where k2 = 2mE/h2. and A = 2rnealnha.

M is here assumed to be finite and single-valued throughout the domain of integration, aa also

(A) is the characteristic integral equation for M-functions of hydrogen; for k3 < 0, it leads to the discrete spectrum, while k2 > 0 : yields the continuous spectrum.

It is easy to transform (A) to the Fredholm type, when k2 < 0 = -a2. We first transform the origin, i.e., put 1-2 = x', etc, in the right side. (A) becomes

(Za+m2+na-k2)M = - hM(x'ylz')dx' dy' dz' . ... (2,32) I [ ( l -~ ' )~+(rn- y1)a+(n-z1)aJ

when ks = -aa, we make a similarity transformation

1 = l'a, x' = ax", etc., and put

N(1, m , n) = M1(1', m', n') wliereby

dx' dy' dz' = a3dx" dy" dz" and

(la+m2+n2+a2) = a2(l'a+m'2+n'2+ l ) , etc.

The relation becomes

A (la+ma+na+l)M = -- 1 M'(x'ylz')dx'd y 'dz' a (Z -~ ' )~+ ( rn -y ' )~+ (n -z ' )a .

If further

then

Integral Equation Associated with Equation for Hydrogen Atom

where

(2,33) is thus seen of the standard form

where the kernel is symmetrical in A and P. Such a transformation will however make the kernel imaginary within the domain, if k2 > 0; to have a uniform treatment to cover both the cases, we wil l not utilise the transformation mentidned above.

We require the following simple result in our subsequent calcula~tions. If r , m d r , arc the distances of a point P(x, g, z) from two fixed points A(n, b , c) and P(f , g, h) , the integral

extended over the whole domain easily transfornls to

in bipolar co-ordinates with A F = k,

and dx dy dz = c3(cosh2[-cos2q) sinh [ sin 7 dCdydq5,

X = cosh 5 and Y = cos 7,

whence, by easy integration,

Again, if

M(u-X, b-y, C-z)dz dy dz M(x'y 'z')dxfdy'dd A(a, b, C) = A 1 x2+ y2+z2 = I (a-~')2+(b-y')2+(c-z')~

. .. (3,2) we have, by multiplying the equation with

and integrating over the whole domain,

A(a b c)da db dc A 1 I ' M(x' y'$)dz' dy'dz' da db dc I [(f-~)Zi-(g-b)~+(A-c)7 [(U-X')~+. ..][(,f-u)~+...]'

258 S N Bass : The Man and His Work

Changing the order of integration and utilising (3,l) we have

henoe, operating with v&,,, on both sides, as

from potential theory, and

v h , (A(f-a, g-b, h-c)} = (vaAW-a, g-b, h-c),

we have finally

M(f, g, h) =- -- ' I (vaA)(,f-a, g-b, h-o) & db do 4+A a a + ~ + c a

ar, a eolution of (3,2).

This important result furnishing a solution of integral equation (3,2) with obvious restriotione about the nature of the funotion A, enables ua to tmokle our present problem.

IV

Aooording to a rell-known theorem due to Hobeon, if an oparator 8 (-& -$ , g) d ia oonstruoted by eubstihuting - .. ., eta for x, y, z in a eolid harmonio 8, of degree

. dx' n (a positive in*),

Let us aseume, with rqeard to the integral eqnetdon (A), that

in view of (4,l) ; aleo (i-P)M(z, y, z) a n be written ee

Integral Bquation Associated with Equation for Hydrogen Atom 259

d d d Sn ( a, 6, E) ~ ( r s )

with the same operitor, as the same surface-harmonic will occur in both. Also as

and

we have

implsins

We have also from ( A )

Henae removing the operator Sn

A[a-x)?+ . . . ]dx d y dz, B = h 1 xs+ ya+z2 9

and hence on account of (3.3)

A = - - - I ( v 2 B ) ( a - x , b - y , c - z ) dx d y dz 4 n4A , x2+ y2+z2

We now perform the integration ; assuming

( a - ~ ) 2 + ( 6 - ~ ) 2 + ( c - z ) 2 = f a = rs+p2-2rp ws O

according to the accompanying figure


we have, as fiif = r p sin 8, m .f-: ;= (p+r )2

d r pB(p2) = inh 1 7 1 3 A ( . f2 ) d f 2

0 f" ( p - r ) '

m dx

= 2 n ~ 1 - - [ ~ , ( p + ~ ) - ~ , ( p - ~ ) 1 , 5

0

where

and G I , an even function of x . Similarly from ( 4 , 5 2 )

where d L .- = v2B(x2 ) ; xdx

hence integrating and putting

we have

where

or Q = x B , and therefore an odd function of x .

Putting finally xB(x2) - Q = G_ -- -

2n2A - 2n2A '

we have the following relations between G , and G2 in the skew-reciprocal form :

0

If the functions G, and G2 are introduced in ( 4 , 4 ) we have


The skew-reciprocal relations (4,53) and (4,541 then a t once suggest that

is true a t the same time.

(4,55) and (4,5G), lead at once, by integration, to

when E2 > 0 ; when k2 = -a2, on the other hand,

0, = - ~ ( x ~ +az)" sin ... (4,58) a

The two equivalent forms of M are

n - - - 27r2A Asn("' --- - - Y, -~ ' ) (-,: ) n [ (r2-k2)

r2 - k2 r sin j A log 1:; it 1 )I . (4,591)

for the case : k2 > 0; while, if k2 = -a2,


v As M is to be single-valued, we see, in the owe of kZ (< 0) = -aa, that a restric-

tion comes in regarding the choice of constants of the problem.

tar l(r /a) is

cos ( 2zh tan-' a n%/a must be an

a multiple-valued function of the form 8+Nn; if therefore r -) and sin ( - a

2nZA tan-' E) are single-valued, i t follows that a a

integer = N. This gives discrete energy-values :

agreeing with the well-known result.

r ? tan-' - = a

Also as

as- r4 fN(rP) 00s ( N 00s-' -- ) = (as+rP)"

aP+ra

we see that M = 0 unless N ) n+l.

Again starting from the series

1 sin N8 = x hN-1 - 1-h cos 8 1-2hcosO+h4 = x hN COS N8, sin 8 ' 1 -2h cos 8+hO ,,

and writing

as-rP COB e = - 2ar

ag+rs and sin 8 = --- aP+r4

we have

ayl-h)+fll+h) - X h N No. {(l-49aa+(l +h)W - ... (6,3)

Multiplying (6,2) with ( ~ ~ + r ' ) ~ - l and differentiating n times with regard to r, we have


... (5'4) from (5,3) we deduce similarly

1-ha ([(1 -h)sa'+(l+ h)zrZln+2 )

Remembering the Gegenbauer expansion

1 .- - 1

lr (2h)' FV (cos 0) - - $ (a2-2ah cos B+ha)" a2"

we can easily deduce, if v = n+ 1 , a = 1 and

the following results :

Comparing (5.4) and (5.5) with (5.6) and (5.7) we have

and

Hence


VI

When ka < 0, the problem. can be transformed to a homogeneous integral equation of the Fredholm type, with a symmetrical kernel

The results that we have obtained above enable us also to say that

are eigen-functions corresponding to the eigen-value -(n + 1 + h)/na for the kernel

Writing

To determine the radial integral, we use (5,7) and write .

which reduces to

where


when, by easy calculation,

Equating coefficient of (ht)x

hence

where Fn(8, q5) is taken to be a normalised spherical harmonio.

We can normalise the eigen-functions of the homogeneous integral equation, for which similarly the value of

will be necessary; for it, we utilise agah (5.7) and write

after easy integration

n (2n+l)! I = -- 1 pn+4 ' -- n! (n+ l ) ! (1-ht)m+a '

Equating coefficient we have

The normalised eigen-functions of the homogeneous equation is


The completeness of the eigen-functions series will mean the equality

(multiple summations as there are repeated roots) leading to the relation

and leading to an expansion of

by a change of axis making (sf y' z') on the z axis

from the well-known addition theorem of Pn(p).

To verify the correotness of the result we note that

Integral Equation Associated with Equation for Hydrogen Atom 267

hence writing

we have

1 - 2 r+rf2- 2rr',ug - ( 1 + r2)(1 +r12)[1 --cosI) cos I)'-sin 0 sin 8' cos $1

4 -V sin NX = T7 ( l + ) ( l + r ' ) sin^ '

if n/d > x > O from (5.1), and cos ,y = oos 8 cos Of+Yi~l 0 sin 0' cos 6; henoe

a result which is a particular case of a general class of identity deduced by Gegenbauer (Whittaker, 1927).

REFERENCES

Elaaswr, (1933), Z e d . f u r . Physik., 81, 334.

Whittaker, (1927), Modern A d y a k , 335, Ex. 42.


Let ters to the Edi tor

GERMANIUM IN SPHALERITE FROM NEPAL

I N VIEW OF THE RECENT USE OF GERMANIUM crystal as a rectifier for ultra-short radio waves, it is worth while to make a thorough investigation of the possible sources of germanium in India. Recently, one of us made a search for germanium in coal ashes from different coalfields in India and a fair concentration of germanium was detected in a few sampleslS2.

Argyrodite, canfieldite and germanite are the only ores rich in germanium known a t present. But none of them have been reported to occur in India. Zinc blende from different countries has been investigated by different workers3 and in majority of samples germanium has been detected in traces. '

Specimens of sphalerite collected from Nepal by one us have been spectroscopically examined. Two of these samples have been found to contain germanium ( FIGS. 1-4 ). The following metals have also been detected in the mineral : Sn, Pb, Mg, Al, Si, Ga, Cd, Ge, Ag and Fe.

During our investigation, we found that germanium occurs concentrated in the magnetic portion of the mineral which seems to be almost pure iron oxide with traces of sulphur,

lead, zinc, gallium and germanium. The non-magnetic portions were also examined spectroscopically and found to contain germanium only in traces.

Germanium was estimated calorimetrically according to the method by Marcel 0rliac4 slightly modified by us.

The percentage of germanium in the magnetic portion has been found to be about 0.24 while it varies in the non- magnetic portion from 0.0072 to 0.04. 300 mg. of pure GeO, has been extracted from 160 gm. of the magnetic portion of the mineral.

The most abundant source of germanium at present in U.S.A. is the concentrate from residues obtained during the melting of zinc ores. Buchanan reports as much as 0.25 per cent of GeO, from these residues. As the magnetic fraction from sphalerite .contains about 0.24 per cent of germanium, it can be regarded as a good source of germanium.

I t appears that germanium has not been detected in any iron ore so far ; thus this new mineral needs further study. We have spectroscopically examined a few haematite and pyrites samples but none of them contains germanium.

We take this opportunity to thank the Nepal Government and the Director of the Nepal Bureau of Mines for the samples of sphalerite. Our grateful thanks are due

The first reference on p. 271 makes a wrong ascription to Dutta and Sen instead of Dutta and Bose, in the original publication itself.

Germanium in Sphalerite 269

F i g .

Fig. 4 FIG. 1 - (a) F E , (b) SPHALERITE. FIG. 2 - (a) F E , (b) MAGNETIC SAMPLE, (c) NON-MAGNETIC

SAbQLE. FIG. 3 - (a) FE, (b) GE WITH TRACE OF F E . FIG. 4 - (a) F E , (b ) GE WITH TRACE OF FE.

to Prof. P. B. Sarkar for his keen interest R E F E R E N C E S during the progress of the work. One of I. MUKHERJEE & DUTT : .%i. G. cwlt., 1948, 14,213. US ( R.K.D.) is indebted to the Council 2. I d e m . : i b i d . , 1949. 14, 538. of ScientGC d Industrial Research for a 3. u R B - 4 1 ~ ~ BLONDEL : Compl. rend., 1910, 150, research grant. 1758.

4. MARCEL ORLAIC : i b i d . , 1945, 221, 500. S. N. BOSE R. K. DATTA

University CoZZege of Science d Technology

Calcutta October 27, 1949

Heprinted from J Sci Indust Res 9B, pp. 52-53, 1950 (Council of Scientific & Industrial Research, New Delhi).

Extraction of Germanium from Sphalerite C o l l e c t e d f r o m N e p a l - P a r t I

R. K. DUTTA & S. N. BOSE University College of Science, Calcutta

A method for the extraction of germanium from the magnetic fraction of the mineral sphalerite has been described. The fraction which is shown to be magnetite contains 0.24 per cent germanium.

I N a previous communication1, we described a magnetic fraction of the mineral sphalerite collected from Nepal. On chemical analysis, this fraction was

found to be mainly ferric oxide with a coating of ferrous sulphide. I t contains germanium to the extent of 0.24 per cent. That the mineral is magnetite has also been confirmed by X-ray analysis. Coal ash, euxenite and some specimens of zinc blende are known to contain germanium in small quantities, but its association with magnetite has not been reported.

E x p e r i m e n t a l

Analysis of the Mineral - The spectrogram revealed the existence of iron, silicon, lead ( trace ) and germanium.

Total Iron - A weighed amount of finely powdered mineral was dissolved in 1 : 1 hydrochloric acid and the total iron estimated by titration with potassium dichromate after reduction with stannous chloride.

Silica was estimated by the usual method, and the value verified by determining the loss in weight by treatment with hydrofluoric acid.

Sulphur - The mineral was decomposed with sodium peroxide and the sulphur estimated as barium sulphate.

Ferrous iron was estimated by decom- posing the mineral with 5 C.C. of concentrated hydrochloric acid and 10 C.C. of hydrofluoric acid in an atmosphere of carbon dioxide, and subsequent titration with potassium dichromate using diphenylamine sulphonate as indicator.

Germanium - Of the various methods known for the estimation of germaniums-5, Orliac's colorimetric method5 with slight modification was found to be the most convenient. A weighed amount of the mineral was carefully roasted a t 500°C. in an electric furnace and transferred to an all-glass distillation apparatus. 40 C.C. of 1 : 1 hydrochloric acid were added and the flask gently heated in a current of air. The distillate was collected in a flask containing dilute caustic soda solution and a drop of phenolphthalein indicator solution. After the distillation was complete, the solution was acidified with dilute hydrochloric acid and the volume made up to 100 C.C. To 10 C.C. of this solution, 1 C.C. ammonium molybdate solution ( 9 per cent ) and 10 C.C. of alkaline potassium stannite solution containing 5 gm. of hydrated stannous chloride and 300 gm. of potassium hydroxide per litre were added and the total volume 'made up to 40 C.C. A series of standard solutions of germanium tetrachloride were

Extraction of Germanium from Sphalerite -Part I 271

FIG. 1 -X-RAY POWDER DIAGRAM OF THE NON-MAGNETIC PORTION FROM THE MINERAL SPHALERITE.

prepared and the same volume of ammonium molybdate and alkaline stannite were added and the total volume in each case made up to 40 C.C. After an interval of 5 min. the transmittancies of the solutions were determined in a Lumetron photo-electric colori- meter, using 440 mp filter, and a standard graph was comtructed. From this graph germanium in the mineral was found to be 0.24 per cent.

The results of the analysis are tabulated below :

Total iron 69.5 F'200, 86.28 Ferrous iron 23.1 27.02 -. Sulphur 1.19 FeS 3.25 Silica 2.16 SiO, 2.16 Germanium 0.24 GeO. 0.34

The mineral is apparently magnetite with a coating of ferrous sulphide. This has now been confirmed by X-ray powder diagram kindly supplied by Mons. Pierre Urbain, Director, Hydrological Institute, Sorbonne, Paris. The rontgenogram is reproduced in Fig. 1.

Extraction of Germanium Dioxide - None of the methods employed by Winkler for the extraction of germanium from Argyrodite was adopted, as the associated elements in this mineral are quite different from those in Argyrodite. After various trials the final procedure adopted was based upon the volatility of germanium tetrachloride and is an elaboration of the procedure described by Buchanan2.

160 grn. of the finely powdered mineral were roasted in an electric mu.%e a t about 550°C. for 12 hr. when all the sulphide was converted to sulphate and oxide. The roast ore was transferred to an all-glass Claisen flask. 400 C.C. of 1 : 1 hydrochloric acid were added and germanium tetrachloride distilled off in a current of hydrochloric acid gas till the volume is reduced to one-

fourth. This distillation was repeated with further additions of hydrochloric acid. A rapid stream of hydrogen sulphide gas was then passed through the distillate for about 15 min. when a milky precipitate of germanium disulphide was obtained ( for complete precipitation, the acid strength was maintained a t 4N to 6N by the addition of concentrated hydrochloric acid ). The mixture was set aside in a stoppered bottle for 48 hr., the sulphide separated by filtration and washed with 6N sulphuric acid saturated with hydrogen sulphide till it was free from chloride. The precipitate on the filter paper was treated with aqueousammonia ( about 8N ) and the filtrate collected. The germanium disulphide was thus obtained as the thio-salt. The filtrate was evaporated, hydrogen peroxide added and again evaporated to dryness. The dried mass was carefully heated to remove ammonium sulphate completely, transferred to a crucible with water, evaporated and finally ignited to germanium dioxide a t 600°C. The yield was 330 mg.

The spectrum of germanium dioxide thus obtained reveals the existence of traces of iron, lead, phosphorus ( impurities from the hydrogen peroxide used ).

A c k n o w l e d g e m e n t

Our grateful thanks are due to Prof. P. B. Sarkar for his keen interest in this investigation, and to the Council of Scientijic Industrial Research for granting a scholarship to one of ,us ( R. K. Dutta ).

R E F E R E N C E S

1. DUTTA & SEN : J . x i . industr. Res., 1950, 9B, 52. 2. BUCHANAN : J . Ind. Eng. Chem., 1916, 8, 585;

1917, 9, 661. 3. TRUDY VSESOYZ KONFERENTSII : Anal. Khim.,

1943, 2, 371 ; C.A., 1945, 3494. 4. HYBBINETTE BL SANDELL: Ind. Eng. Chem.,

Anal. Ed., 1942 14, 715. 5. ORLIAC : Comfit. Rend., 1945. 221. 500.

~ e ~ r i n t e d from J Sri lndust Res SB, pp. 251-252, 1950 (Council of scientific & ~ndustrial~~esearch, New

Delhi).

Extraction of Germanium from Sphalerite C o l l e c t e d f r o m N e p a l - P a r t I 1

R. K. DUTTA & S. N. BOSE University College of Science, Calcutta

A method for the extraction of germanium from the non-magnetic fraction of the mineral sphalerite has been described.

I N the previous communication1 we have described the extraction of germanium dioxide from the magnetic fraction ( mainly magnetite ) of sphalerite. The

non-magnetic portion left over was found to contain traces (0.006-0.04 per cent) of germanium.

The spectrogram of this portion revealed the presence of the following elements : lead, zinc, iron, silicon, germanium, tin, copper, cadmium, silver and antimony, of which the first two predominate. The ore was analysed as before and the germanium estimated according to the method already described.

The analytical results are as follows :

Pb %

14.2 Zn 48.4 S 27.3 SiO, 5.6 Fe 2-3 Ge 0.0021

From the analytical data, this fraction of the mineral appears to be mainly a sulphate of zinc and lead.

Extraction of Germanium

A procedure different from the one adopted for magnetite had to be adopted for the extraction of germanium from this fraction. When the ore is treated with sulphuric acid the insoluble lead sulphate formed retains the major portion of the germanium but if the sulphuric acid is completely fumed off, most of the germanium goes into solution and only traces remain over with the preci- pitates of Iead sulphate and sulphur.

The methods of Winkler, Buchanana, etc., werc found to be unsuitable for the extraction of germanium. Tchakirian's methods was finally adopted. The insoluble residue was re- fluxed with a saturated solution of ammonium

bioxalate and, curiously enough, not even a trace of germanium was found in the solution. A new method based on the fact that the insoluble residue containing free sulphur is totally converted to lead sulphide on treatment with concentrated sodium hydroxide solution was adopted. The dismutation of free sulphur in strongly alkaline medium takes place as follows :

4s + 60H = 2 5 - + Sa08 + 3H,O The lead sulphide so formed carried with it the whole of germatium as disulphide and the lead sulphide is separated by treatment with 2N hydrochloric acid.

E x p e r i m e n t a l

400 gm. of the ore were decomposed by an excess of 1 : I sulphuric acid, the resulting mixture heated to dryness and the excess acid fumed off. A large part of the free sulphur was removed by this treatment. The dried mass was extracted with hot water and filtered. Sodium sulphide, just sufficient to precipitate one-tenth of the total zinc, was added4. The filtrate was rejected as spectroscopic examination showed the absence of germanium. The residue was treated with 4N sulphuric acid and digested on a water bath for an hour and filtered. Germanium disulphide was left in the residue and most of the elements including zinc were thus removed. The residue was treated with sulphuric and nitric acids and dried. I t was then distilled with 50 C.C. of 1 : 1 hydrochloric acid and the distillate treated according to the procedure already described. Germanium dioxide thus obtained was found from spectroscopic evidence to contain traces of antimony and tin.

When the residue was treated with concentrated sodium hydroxide solution, the whole of germanium was precipitated along with lead sulphide. This precipitate was digested with 2N hydrochloric acid, filtered and washed. Almost the whole of lead goes into solution leaving behind germanium

Extraction of Germanium from Sphalerite -Part ZI 273

disulphide, silicon dioxide, titanium, silver and traces of lead. Silicon dioxide was the principal constituent. All the vapours issu- ing during the treatment were collected and found to be spectroscopically free from germanium showing that germanium disulphide is not decomposed by 2N hydrochloric acid. The residue was then treated with sulphuric and hydrofluoric acids and evaporated on a water bath, when all the silica was removed. The mass was then distilled with hydrochloric acid, as previously described, and the germanium dioxide separated. The resulting product contained traces of phosphorus derived from hydrogen peroxide.

The yield was 240 rng. of germanium dioxide from 4 kg. of the non-magnetic fraction.

Our best thanks are due to Prof. P. B. Sarkar for his keen interest in the investigation and to the Council of ScientiEc 6 Indus- trial Research for financial help to one of us (R. K. Dutta).

R E F E R E N C E S

1. DUTTA, R. K. & BOSE, S. N. : J. S C ~ . industv. Res., 1950, 9B, 251.

2. BUCHANAN : J . Znd. Eng. Chem., 1916, 8, 585: 1917, 9, 661.

3. TCHAKIRIAN : Ann. Chim., 1939, 12, 415. 4. URBAIN et al. : Compt. Rend., 1910, T 150, 1758.

Reprinted from J Sci Znduet Res SB, pp. 271-272,1950 (Council of Scientific & Industrial Research, New Delhi).

PHY sr QUE T I ~ E O R I Q U E . - Ides identitds de divergence dans la nouvelle

theorie unitaire. Note de M. S . N. BOSE, presentde par M. Louis de Broglie.

Einstein (') a deduit les quatre iden tites: de divergence de sa nouvelle theorie des identiles de Bianclji, en y levant l'ambigu'i~e de la differenciation covariante.

La mdthode de Hilberl ( 2 ) leghrement modifihc fournit nussi immddiatement le resultat; il est peut-&re interessant d'indiquer le mode de calcul.

La relation entre les coefficients de connexion affine r", et gpv n'est pas simple dans la nouvelle thdorie. On ne peut donc appliquer direclement ici la melhode de Hilbert de variation d'une fonction invariante de pV et de ses dCriv6es. Dans la nouvelle theorie, on ne suppose pas inilialement cette relation e n t r e r et g et la variation donne les equations ainsi que les equations de connexion de I' et g. La nouvelle d6monstration proposde tient compte de ceci. On comprendra la methode d'apres les indications suivantes.

Forrnons le tenseur I&,,,, contractti du te6seur de Riemann gdnkralisd. Son caractere covariant est evident d'apr8s la loi de transformalion de r:,, comme dans laihdorie symdtrique. Y R , est donc un invariant dans les changemen~s de coordonnees, de meme que l'expression I gItndx, dx, dx, dx, = I g ('"dv,.

Si R,, = o l'in legrale

pour une variation quelconque des seuls p; H,, etant fonction des coel'ficients de connexion affine n'est pas change dans la variation.

Observons avec Hilbert ( ¶ ) que

construit h partir d'un vecteur contrevariant arbitraire rfo p u t Ctre p r i ~ pour 8gpv dans (I). D'OU

( I . ) Canad. J . Alath., 2, 1950, p. 120 et suiv. (') Math. .4nn., 92, 1924, p. I et suiv.

Les Identitks de Divergence duns la Nouuelle Thkorie Unitaire 275

L'integration est etendue a une mul~iplicite a qualre dimensions limitee par une mul~iplicitd a trois dimensions et l'on a partout R,,= n.

Nous observons que

Si nous considCrons pa= A x n comme une variation convenable pe t i~e et arbitraire des coordonndes, variation qui s'anqule h la frontibre, nous obse~.vons

-- et par consCquent = o, puisque \/g-dv, ert invariant.

En mul~ipliant ( 4 ) par dv, et . en integrant sur la multiplicilC a quatre dimensions on a

= une integrale de surface = o , d'aprks l'hypolhbse de la variation. On tire donc immddiatement de (3) el (6)

identiques au resuhat d'Einstein (=). Toutefois, d'une certaine facon, notre resultat est 1CgCrement plus general. On peut -dire que le raisonnement d'Einstein est une me~hode algebrique, ou les connexions enire les coefficients affines et les p sont admises apriori, ainsi que les condi~ions d'in~egrabilite et l ' l ~~po thkse particulibre I'&- I';, = o.

Notre dkrnonstra~ion esl independante de ccs hypolhbses restrictives. En fait, en essayant de ddduire les equalions generales R,,= o et les dquations de

connexion du principe de varia~ion 2 [@"R~, ~ g d v , = o, on cst conduit a la - connexion d ' h s t e i n sculetnent si I':, - 1':,= o.


D'autre part, en posant aph'ori la connexion dlEinstein, la compatibilite dc la variation avec l'hypothkse initiale conduit a r, = r:, - Tg', = O .

Puisque la demonstration donnee ici n'u~ilise pas les dcpatlons de co'nnexion. elle montre donc nettement que les identiles ci-dessus sont independantes des equations de connexion. En fait, les k,, sont fonctions des r seules, et la ddmonstra\ion des idenkites de divergence suppose seulement I'existence des derivees premieres des g". Elle subsistent mCme quand par suite des singula- rites du champ des @", il n'existe pas de derivees secontles 011 d'equa~ion de connexion integrable.

Reprint4d from Comptur rendus de I'Acadernie den Science8 41)6, pp. 1333-1336, 1963 (Gauthier - Villars Pubiishers, Paris).

The Identities of Divergence in the New Unitary Theory

Note of Mr. S. N. Bose, presented by Mr. Louis de Broglie

in stein(') deduced the four identities of divergence from his new theory of Bianchi identities, clearing the ambiguity of covariant differentiation.

The method of ~ i l b e r t ( ~ ) , when slightly modified, also immediately gives the result. Perhaps it is interesting to indicate the mode of calculation.

The relation between the coefficients of affine connection r F, and g" is not simple in the new theory. Thus the method of Hilbert regarding the variation of ap invariant function of g" and its derivatives cannot be directly applied here. In the new theory, this relationship between r andg is initially not assumed and the variation gives rise to the equations as well as the equations of connection of r and g. The proposed new demonstration takes this into account. The method will be understood according to the following indications.

Let us form the tensor R,,, contracted from the generalized Riemann tensor. Its

covariant character is evident from the law of transformation of r ;, as in the syrnmet-

ric theory. gp R,. is' thus an invariant under changes of the coordinates, as in the expression Ig 1 ' I 2 dxl dx2 dx3 dx4 = Ig 1 lJ2 dud.

If R, , = 0 the integral

for any variation of gp ' alone ; R,, being a function of the coefficients of affine connection is not changed in the variation. Let us observe with Hilbert ( 2 ) that

obtained from an arbitrary contravariant vector cpa can be taken for 6gwv in (1). Hence

The integration is extended over a manifold of four dimensions limited by a manifold of three dimensions and one has everywhere R, ,, = 0. We observe that

- --

(I) Canad. J. Math., a, 1950, p. 120 et seq. ('1 Moth. Ann. B2, 1924, p. 1 et seq.


If we consider cp" = A xaas a suitable small and arbitrary variation of the coordinates which vanishes on the boundary, we observe that

ad9 !@ az. dv4 = MU, and - - dx" y a = Adg. 1 and 1

and consequently = 0, since G d u 4 is invariant. By multiplying (4) with du4 and integrating on the manifold with four dimensions we have

= a surface integral = 0, as per the hypothesis of variation. We immediately obtain from (3) and (6)

A partial integration for eliminating the derivatives of Q" immediately gives the identities

gm dg --- axe dzy W' Bpi +gyp Rap 11/6 = 0

identical to the result of Einstein (3). Nevertheless, in a certain way our result is slightly more general. We can say that the reasoning of Einstein is an algebraic method where the connections between the affine coefficients and the gV as well as the conditions of integrability and a particular hypothesis r :, - r ;a = 0 are assumed a

priori. Our demonstration is independent of these restrictive hypotheses. In fact, while

trying to deduce the general equations R,, , = 0 and the equations of connection from

the principle of variation 6 J g" R,, , du4.= 0, we are led to the connection of

Einstein only if ca - a = 0. On the other hand, by assumingapriori the connection of Einstein, the compatibility

of the variation with the initial hypothesis leads to r, = a - T: , = 0. Since the demonstration given here does not use the equations of connection, i t

(') C d . J. Mdh., 2, 195O.p. 126, equation (176).

The Identities of Divergence in the New Unified Theory 279

clearly shows that the above identities are independent of the equations of connection. In fact, the R,, are functions of r alone, and the demonstration of the identities of divergence only assumes the existence of the fvst derivatives of gp ". I t survives even when due to singularities of the field of g"", there are no second derivatives or integrable equations of connection.

Ewlish translation of Compteo rend- de 1'Accrdcrnie dc. Scienca W, pp. 1333-1336,1963 (Gauthier - V i h m Publishers, Paris).

UNE THBORIE DU CHAMP URITAIRE AVEC I+, j o

Par S. N. BOSE. Univanitd da Calcutta

SO&. - On Ctabllt dea CQuPtlona du champ uuitaim en hisant varier une intCgrale qai sotidpit au postulat d'hmnitidk On obtknt 1 ' ~ ~ e s form& avcc lea thbries andamem, wit ~ V C C r; - 0, soit avec 11 a+ 11 - o quu~d 0, - 0 ; on pent Intm'prCter (4 comme la fonr Q Lorentr ri on p m d lea a+ comme le~ composma du champ Llaetrom(~Ltiqu8.

Les Lquatiour approebbs avec compoaantea oaUsymLbiq~c. prsnnent la formc maxasllienne pour une certaine vrlaur dw e o ~ t e s , Toutefoir, pour me autm valeur, il cst posaibls d'avoir -k ehPmp.ronr aucaw aingukita.

On p u t classer en trois g r o u p lee Bquations caracthristiques de la thhrie du champ d6Einstein :

I. Lea Bquations du6 ohamp qui inthreasent uniquement les coefficients de connexion I-':, et leura derivw.

11. Iss Bquationa de connexion qui wlient le tenaeur g aux coefficients de oonnexion a tbe .

111. Lea quatre relations de 0 ~ f 8 ~ t B r e mtrictif :

Notons que lea relations (111) eont semblablea aux 24 Bquations

r;#-r;,, = 0.

qui, dans la thhrie wlativiete de la gravitation sont les condquenm du postulat de la symm6trie de la connexion affine.

Dens la thbrie du champ unitaite aussi bien que dans la thhrie de la gravits- tion, les Bquations due champ (I) et les 6quations de connexion (11) peuvent se deduire simulC(u16ment d'un principe de variation oh l'on dgale B zero la variation arbitraire d'une int&ale, la variation Btsnt cepedant mujettie aux restrictions du type (111).

N6anmoins les conditions arbitraries sont bien moins nombreuses dans la thhrie unitaire que dam la thbrie de la gravitation.

Dans cet article, on essayed'6liminer toutes les oonditions restrictivee et d'Bteblir un systkme d'equations fondamentales en s'appuyant uniquement sur 10 prinoipe de variation.

On a toutefois utilW le postulat d'hermiticiu rtkemment dnoncd par Einstein pour oonstruire I'int4grale dont la variation conduit B cette nouvelle gBndralieet~on.

On v e m dam lea rh~ l t a t e pr6cia8s ci-dessous - que des terrnea additionnels con- tenant r,, interviement aussi bien dens les equations du champ que dans leqequations de comexion.

Une Thborie du Champ Unitaire avce TN # 0

Lea Bquations de connexion qui concernent le tenaeur g donnent lieu B quelquee remarques. Ellee preneent iai la forme

&"+Pa ria+9.r = 3 8 f i V @ h ,

tan& que dens les deux theories pdddentes le premier membre de cette 6galiM eat nul.

4, eat defini iai par lea Bquatione suivantes :

1 *, - '' avw aAe = 5 (gAL -LA ) ' - Ql-ifl'

Si I?, = 0, k8 = 0 et par consequent on a aurtsi @ = 0 et 1'6quation reprend la forme ancienne.

Ndanmoins il existe une autre possibilite : que ass soit la partie antisym6trique du tenseur covariant consider& comme l i b au vecteur & six composantes (E, H) de lrt theorie BlectromagnCtique.

Si le determinant

I I a ~ n Il = (a12a34+a31a24+a23a14)' = 0,

on peut avoir 0, = 0 m&me si k g et r, ne sont pas nuls.

Remarquons que pnisqu'on suppose toujours

I I sBu+si" I 1 # 0,

kfi et r, sont nuls simultanem6nt.

La condition/[ a ,, I( = 0 combinb avec la rbgle hrtbituelle de corr6lation citee ci-deasus conduit immediatement & (EH) = 0, propriete clbsique du champ Blectro- magnbtique.

On est anisi tent6 de poser une correlation d'une part entre kC et le veoteur quatre dimensions courant-charge (car k: = 0 dbcoule immediatment des equations g6n6rales) et, d'autre part, entre @, et la force ponderornotrice de Lorentz, en raison de l'identification classique

On peut interpr6ter @, = 0 comme d6terminant une distribution stationnaire ou la force de Lorentz est nulle, ce qui r a m h e l'equation de connexion & la forme qu'elle evait dans les theories pdc6dentes.

l h n s les Bqustions tle champ, 1'6lement symBtrique Pi,, e t llB1bment antis-

yln6rtrique Q,;, ont cles roles essentiellement diffbrents.

Au moyen d'hypothbms sur les ordres de grandeurs relatifs des diffbrents t e r mes, nous avons pu ramener les'equations fondamentales & une forme simple.

Physiquement, ces hypothbses impliquent qu'il est possible de nBgliger la gravitation dans 1'Ctude des relations entre grandeurs Bleotromagnbtiques.


TAB Bquations sinsi transform& ont une forme interessante e t l'on peut lea inter- pdter comme de simples condquences de la thbr ie de Maxwell si l'on attribue oer- taines valeurs partiaulihres aux constantas a ~ b i t F w k . Ma$ il est possible aussi d'obtenir un champ sans singularit6s pour d'autres veleurs partiouli6reg. Le ohamp n'est alors Bvidemment pas msxwellien.

Ces aspects encourageants du nouveeu systltme d16quations m'ont amen6 8. penser qu'il conveneit de publier les premiers r6sultats provoquer les oritiques e t lea obwrvations.

Suivant Einstein, tolltea les relations de la thBoire unitaire doivent satisfaire nu po~tdfbt d'kmibicit6, c'eat-k-dire rester invaricmtes q u a d gbr',g,,,, r:,,, sontc hang&

I In fois rn # I 1 , g,,, , r,', par permutation dcs indices / r e t % J .

On vo-it facilement qrlc clans cettc transformation le tenseur d ' E i n s t 6 E de oompsan t es

El,, = l';l,,h - qh , 1- q, qA - rf4 r;, derient le tenwwr 11 de composantes

HI,, = r;,,A-r;,,r+r;,,, rir-r:; T ; ~ . Au lien d~ l'exprewion

I = 81". E , , d 1 g 7 g'C1' E,,,. qui intrrrient dans 1'intbgra.le sol~misr ii ~nriation poiir ohtenir les Bqitatioru fondamentales, now doptcms mr forme plns g6n6rale

pnispiie la permutation de 11 et v ne modifie pas lc d6terminant. Les termes additionnels vCrifient aussi la condition tl'hermiticitt! pnisqw r,Aflr' ahangent de signe et que yfl" conserve 1e sien arcc la ~>ermut,at,ion.

S6parons dms les coefficients de conncsion ~ffine. la partie sym6trique Pi,, r t la part,ie mtisymktxique I/:,, rt posons encotc

Une Thdorie du Champ Unitaire avce Tc( t 0

8 V e 0

r, = v:,, on e

@*= 0.

En effectuent les substitutionx clans Z', tous lee termes qui mnt produita d'un facteur sym6trique par un fncteur sntisym6trique dispmaissent dam la sommation et nous aVom findement

oti x, y sont des constante arbitraires,

est la comp0ssnt.e sym4triA (111 tenseur #Einstein ayec connexion affine aym6- trique et

= % - G,PL - eve, + a;$% est la d&iv& covariante claaeiciue cdcul4e auwi avec connexion sym6trique.

Au cows de la variation arbitraire de

L = $ Ifdv4, dv, = dz%IxW&. yNu, A'"' conservent automatiquement leu1 caractere sydtrique. Lea variations de yFY et AFu puvent donc etre considdmh comme arbitraires, ae qui donne iaam6- diatement lea Bquations du champ

arf - = E,,. - a;,@,, .+ x r r r , = 0, ayr'

a l f A a j , ~ = Q F u ; h - ~ ( r r , r - r v , p ) .

Pour obtenir lea 6quations de coneexion noue Bgalons d'abord I' B une divergeno8 B quatre dimensions (H) et considerons H seulement puiaque la divergence se trans- forme en une integrale Btendue & un domcline B trois dimencriom oh tous lea coeffioients des variations arbitraires s'annulent.

Rappelom-nous cependant que dans la v,ariation des 616ments F,, Pi,., Gb,, les qua* relations @LA = 0 rmtent toujours valables; lea 24 oomposanta de ne eont pas arbitrah. Nous devona donc ~ppliquer la mithode habituelle, employer dm multiplicateurs indi:terminb kfi et varier la fonction

H-2kfiOtA = H'.

&a coefficienb k# seront d6termines en demier lieu B partir de 1'6quation finale. NOUS obtenons Ainei par un calcul facile le syst6me d'6quations

f< + yY8P;, + f " P L - y"'Pf, = - [ A W L + AavC7&'J,

284 S N Baw : Tha Man and His Work

D'oh on tire imddbfenmnt

b : - 0

Now ponvons maintenant donner AUX &pations Cle, oonnexion la forme oon venrble : par Rddition et mgmupament on a

q;Lr + g'mq, + gfULf, - g'rrL;, = k*U:-Wf,

oil 4. = Q:'+P:, ,

en mnltipliant par 1- coefficients g',, d6Anie par

pub en remplmnb L:, par sa vdeur et en divisant par %/c. nous obtenons apr& uw, tmmformation f~oile

En obtwrvant que

Lei nouveeur am6ffioienta atiinea aont

Une Theorie du Champ Unitaire aoce TI # 0

En introduisant oes nouveaux coefficientv affines dans 1'6quation fondamentale, nous pouvons dcrire

t b,a(g'"+g"") = 8:.

3. CAS PARTICULAR

Quand Ox = 0, mais r,, k@ f 0, on a

a,fi,+~~+(c31a2, = 0.

Effectuons la transformation et observons que r; = Mr, ;

Pt , et Gi, ne ohangent pas et nous obtenous

&, +tr;r;-qC G:, = 0;

(Q:,);A + t(ry,r-ru.rr) = 0, la eeule modifioation est une Muction du nombre des consbantea arbitrairea.

LYBquation de connexion est (puisque @A = 0)

gfA + par; , + g' rtx = 0. mais evec r, # 0.

On montre dors failenlent que

(Ar'db)Y = f (#'a + gW..

Nous supposons que lea coefficients antivy~nBtricluex A"" st Q;, wnt ptita, voiains de m o .

Alors les Pi, peuvent auasi Btre n@lig&,

yNV = 8; ,

la &'vation oovariante est remplwtBe par la derivation ordineiw, et now avom findement en pr : m i h approximation

S N Bose : The M a n and His Work

e t enlin cornme

En 6liminant k' nous obtenons

Si 1 = 2 nous pouvens poser 28

A:' + AzA + A? = 0

et 1'8quation devient identique aux dquatione du ohemp BleotwmagnBtique ds Maxwell.

1 * --.!!.- 2 8 - 0, lea dquatioas deviennent

uptttme different Bvidemment de ~ l u i de Wwell , msie qui p u t avoir une solution sans singulariti8s en auceun point.

11 eat peut Ftre intkressant de remerquer que certe,ine6 mher0hee la q w t i - fication du champ dleotromagnetique ont emend de leur dt.4 B p- A?,' 0, 00 qui entrainsit implioitement une modifioation des 6quatione due ohamp.

Si I1 a,, 11 = 0 on ohtient une approximetion p rhn tan t lee m6mee o ~ ~ ~ *

Reprinted &om L. Jour de Phy8 et k M u m (Paris) 14, pp. 641-644 1963.

A Unitary Field Theory with I?, # 0

By S . N . BOSE, University of Calcutta

Summary - The equations of the unitary field are established by varying an integral which satisfies the postulate of hermiticity. One obtains an absolute equivalence with the old theories either with T,, = 0 or with 11 a 11 = 0 when c4, = 0 ; one can interpret as the Lorentz force if ak, are taken as the components of the electromagnetic field.

The approximate equations with antisymmetric components take the Maxwellian form for a certain value of the constants. Nevertheless, for another value, it is possible to have a field without any singularity.

1. Introduction The characteristic equations of Einsteiri's field theory can be classified into three groups :

1. The field equations which involve only the connection coefficients l$, and their derivatives.

11. The connection equations which link the tensor g to the affine connection coefficients.

111. The four relations of restrictive character

qA-qp = 2r,' = 0.

Note that the relations in (111) resemble the 24 equations,

r;,-r;, = o. which, in the relativistic theory of gravitation, are consequences of the postulate of symmetry of the affine connections.

In unitary field theory as well as in the theory of gravitation the equations of the field (I) and the equations of connection (11) can be simultaneously deduced from the principle of variation where the arbitrary variation of an integral is equal to zera, the variation being subjected t o restrictions of type (111).

Nevertheless, the arbitrary conditions are much less numerous in the unitary theory than in the theory of gravitation.

In this article we are trying t o eliminate all the restrictive conditions and establish a system of fundamental equations by relying only on the principle of variation.

We have used the postulate of hermiticity recently enunciated by Einstein to construct the integral, the variation of which leads t o this new generalization.

The specific results given below will show that the additional terms containing T, occur as well in the field equations as in the equations of connection.

The equations of connection which concern the tensorg give rise to certain remarks. Here they take the form


whereas in the two proceeding theories the first member of this equality is zero. Here QL is defined by the following equation

and

If r,, = 0, kB = 0 and consequently if we also have 0 = 0, the equation again takes the old form.

However, there exits another possibility : that a, B being the antisymmetric part of the covariant tensor, is considered as linked to a vector with the six components (E, H) of the electromagnetic theory.

If the determinant 11 ah# 11 = (a,fis+as1%+a&l4)' = 0,

we can have QI, = 0 even if kP and rP are not zero. Note that since we always suppose

II sBv+s* I l f 0, k' and r, are simultaneously zero.

The condition ]I aI, 11 = 0 combined with the usual rule of correlation just mentioned immediately 1eads.b (E H) = 0, a classical property of the electromagnetic field.

One is also tempted to put a correlation on the one hand between k' and the four dimensional charge current vector (because k$ = 0 comes immediately from the general equations) and on the other hand, between al, and the ponderomotive force of Lorentz, due to the classical identilkations

We can interpret a1 = 0 as determining a stationary distribution, where the Lorentz force is zero. This brings back the equation of connection to the form which was found in previous theories.

In the field equations the symmetric element eV and the antisymmetric element

G 3 have totally different roles. By means of hypotheses on the orders of relative magnitude of the various terms we

could bring back the fundamental equations to a simple form. Physically, these hypotheses imply that i t is possible to neglect gravitation in a study

of relations between electromagnetic quantities. The equations thus transformed have an interesting form and we can interpret them

A Unitary Field Theory with TN # 0 289

as simple consequences of the Maxwell theory if we attribute particular values to the arbitrary constants.

But i t is also possible to obtain a field without any singularities for other particular values. The field is then not Maxwellian.

These encouraging aspects from the new system of equations have made me think that i t would be appropriate to publish the first findings in order to provoke criticisms and observations.

2. The new equations Following Einstein, all the relations of unitary field theory must satisfy the postulate of hermiticity, i.e. remain unchanged when gp, g, ., are changed to gV ", gv ,, , e, all together through permutations of the indices p and v.

We can easily see that in this transformation, the Einstein tensor E of components

which occurs in the integral which is subjected to variation in order to obtain the fundamental equations, we adopt a form which is more general

1 1 where ,,w = -(p+p) = g ,/ m- (p+q? 2

since a permutation of p and v does not modify the determinant. The additional terms also satisfy the cqndition of hermicity since T, A' ' changes sign and ' retains its sign with permutation. Let us separate the symmetric part eV and the antisymmetric part e, in the *ne connection coefficients $ and let us also put

with


one has 0.

Through substitution in 1', all the terms that are products of the symmetric factor and the antisymmetric factor disappear in the summation, and we finally have

where x, y are arbitrary constants. 1

E,,. = P:l,,,-T(P:A,r,+P;h,r)+P:v P $ h - I ) : h P:" is the symmetrized component of the Einstein tensor with symmetric affine connection and

G;l,;A = G;l,,A - GicPi , , - G$l,PiA + Gi,,P:A

is the classical covariant derivative, also calculated with symmetric connection. During the course of arbitrary variation of

L'= $ I'dv,, dv , = dx1dz2dx=d.d.

f ' , A'" automatically retain their symmetry character. The variations of $' ' and Ap' can thus be considered as arbitrary, which immediately gives rise to the field equations

-- 'I' - E,,,, - ~; ,a: , + ~ r , r , , = o, dyfir'

In order to obtain the equations of connection, let us first make I' equal to a divergence in four dimensions (HI and let us consider 6nly H since the divergence transforms itself to an integral over a three dimensional domain where all the coeffl- cients of arbitrary variations cancel each other.

Let us recall the fact that in the variation of the elements T, , piv , G; the four relations G; = 0 always remain valid ; the 24 component of G;. are not arbitrary. We can thus apply the usual method of employing undermined multipliers kW and varying the function

These coefficients kp will be determined in the end from the final equa6ion. Thus through easy calculation we obtain a system of equations

A Unitary Field Theory with Tcl P' 0

and consequently A fZ' = 3kfi,

From where we immediately obtain

k; = 0

and 5

kfi = .6yfiY I"", aveo 6 = - - . 3~

We can now give a suitable form to the equations of connection : through addition and regrouping we have

where

by multiplying with the coefficient g',, ,, defined by

we first obtain L*: = a ' - C7 A + a t A B k ~ ,

C

and

then, by replacing Lz a by its value and dividing i t by 6, we get An easy transformation

and by regrouping and adding certain terms, we finally obtain

The new affine coefficients are

292 S N Base : The Mcuc and His Work

and

By introducing these new affine coefficients in the fundamental equation, we can write

3. Particular case

When @I = 0 but r,, , k # 0 we have

Let us make the transformation and observe that

e; and G; ,, do not chnge and we obtain

the only modification is a reduction of the number of arbitrary constante. The equation of connection is (since = 0)

but wi,th r, it 0. Thus it is easily shown that

(A"4&), = * (BC. + QOCV'.* 4. Approximation

In order to find an approximation we put the equation of connection in the form

A Unitary Field Theory with # 0

Let us suppose that the antisymmetric coefficients A' ' and G; ,are small, close to zero.

Thus PI,, can also be neglected :

yrr =a;, the covariant derivative is replaced by an ordinary derivative, and we finally obtain in the first approximation

A? - k"8: + b 8 f = -[q, f G$],

A:' = 3b, from which we can easily deduce

(a:,) = A?-- k"8l + kY8tf -) [Ay + AAf: + A:'],

and finally with

kr r' = 8, we obtain the following system

1 3Y A:: - -[A:' + A:' + A?''], = [k: - k'&]. 4

By eliminating kw we obtain

1 1 A:;( I - $ ) - ~ ( -$) (A:. + AiA + Ayh = 0.

1f 1 = L we can put 2 8

and the equation becomes identical to Maxwell's equations of electromagnetic fields. 1 But if z - = 0, the equations become

9

with


the system being evidently different from that of Maxwell, but which can have a solution without singularities a t any point.

I t is perhaps interesting to note that certain researches on the quantization of the electromagnetic field have put the subsidiary condition Af: = 0, which would imply a modification of field equations.

If 11 a k, 11 = 0, we obtain an approximation presenting the same characteristics.

Manuscript received 18 July 1953.

English translation of Le Jour de Phys et le Radium (Paris) 14, pp. 641-644, 1953.

CERTAINES CONS@UENCES DE L'EXISTEHCE DO TENSEUR g DANS LE CHAMP' AFFINE RELATIVISTE

Par S. N. BOSE, Univenitl de Calcutta.

Somnulrr. - On Ctnbllt dm relations oh antrent rulement I$., et bun dCrlvCes P pPrtlr des condl- tions d'tntlgmbUtC des &pations anxquelles utirfdt Is t4nrear g dm thCories mlativlates.

Le nombm de relations indlpandantes entn ClCments du champ est bien plua grand an thlorb nnitaire qu'en thCorie de la gravitation.

1. INTRODUCTION

Le formalisme abstrait de la' th6orie de la relativitk a actuellement abouti t i la conception d'un champ de coefficients do connexion affine d'une vari6t6 4-dimension- nelle.

Soient, 8x' les composantes d'un dkplacement parallele e t infiniment petit auquel un vecteur A de la vari6M consid6r6e est assujetti: les variations des composantes A" qui en r6sultent sont donn6es par la rhgle

SA" = - An PiB SX* ... ( 1 . 1 )

Dans la thborie de la gravitation les termes I'Z8 sont sym6triques, ou I'$ = rl;, ; donc, la loi de transport y est unique.

Au contraire, comme les coefficients sont dissym6triques d a n ~ la tl~korie unitaire, on p u t y avoir aussi une deuxibme loi de transport, savoir :

En s'appuyant sur la rele (1.1), on peut calculer facilement le tenseur dc courbure R dont les composantes

efi = r&.v - riw.fi + 'LrR- rLr2fi I I

sont antisym6triques en indices p et u : et ensuite, par voie de contraction, on a facilement le tenseur d'Einstein E, avec les composantes

E.W = q,',* - I' ik, , ' + q K I ' ; & - I'$'$' (1.4)

Mais la th6orie unitaire adment aussi (1.2) comine loi du transport; donc. ie meme raisonnement que ci-dessus nous donnera, A partire de (1.2) un autre transport S (substitut6 b R) avec des composantes

Sin ca, - Ca, fl + I'iaCC - I';raT'tt (1.5)


e t findement aussi le tensem H (substitute & E Einstein) nvec des composantes

H,', = rim,* - rL,, F r';,G - riac. (1.6)

Les Bquations E,, = 0 sont celles du champ dans la thBori de la grevitation.

L'ambiguitk de la loi du trtmsport dans la thBorie unitaire opposeapparement des difficulties au choix de E,, = 0 comme Bquations fonndamentales.

Mais celles-ci ont Bt.4 tournbs par Einstein de la maslibre suivante : il impose au champ affine quatre conditions restrictives snvoir

rbA - r:, = zr, = 0. (1.7)

Bein qu'arbitraire, mtte h y p o t h h donne facilement

E = H ;

lea deux rbgles conduisent ainsi au &me tenseur; donc lea equations du champ peuvent

rester lea &mes dam Ids deux thtbries.

A &tk du systeme

E,,. = 0

qui depend seulement des coefficients Pi, et de leurs dBriv&s premieres, il y a aussi un second systhme dans lequel entrent Bgalement les composantes du potentiel g@" et leurs dBrivQs : ce sont en effet les Bquations

gy + gpAriA + gAYr& = 0. (B)

On a dBmontrB que (A) et (B) se laissent dBduire d'un principe variationnel.

On forme I'integrale invariante

avec des composantes du potentiel e t du tenseur d'Einstein (suppos6 unique B cause de r, = 0) e t l'on pose

81 = 0

pour toutes variations arbitraries de g/41, e t I?:,, ; on obtient immediatement les Bqua- tions cherchbes.

I1 est Bvient que l'existence du tenseur de potentiel dans le champ affine donne- rait & celui-ci un caractere tout & fait pnrticulier.

En effet (B), regardhe comme un s y s t h e d'6qnations diffkrentielles, permettra de dBterminer grr' seulenlent quand les conditions d'int6grabilitB seront satisfaites.

Ainsi, les relations qui suivent colnme cons6quenccs tle cette hypothBses d'intB- grabilite mtre les coefficients r;, et Ieur, cli.ri\-8c~ e\prinwrontcelles des propriBtBs carect6ristiq11es du champ, qui sont postulCes implic~te~nent par 1" thBorie de la relatirite coinme par la thCorie actuellc tln chn~np unitaire.

Certaines Conskquences 1' Existence du Tenseur g

Les obtenir sous une forme explicite est prkcishment le but de ce travail.

2. CAE I. Quand les coefficients I'i, sont symktriques, les conditions d'inGgrailit8 des Bquations (B) savoir :

g,gY + g u x r i A + gAvr:k = o

8ont plus facilement exprimables par l'intermhdiaire du te~lseur de courbure R.

Si les composa~ltes RI, soht cxprini6es par les multiples gfiu et des parenthhses n

ir quatre indices comme

Ri',,, = gAo (klmn), I

les conditions d'intkgrabilith seront bien exprimhs par les propri6Gs suivantes et bien wnnues de ces patenthhes, savoir :

(klmn) = - (lkmn) = (mnkl) = - (nmkl). ... (2.2)

A partir de celles-ci, on p u t %miner simultanBment gPw et les parenthhs et obtenir des relations entre leu composantea de R seulement.

(i) Comme

on pose h = 2 et l'on contracte. On observe que les p a r e n t h h sont antisym6triques

en indim (k et 2) B st6 de dk qui sont symBtriques.

On e alora automatiquement :

(ii) De mGme, oomme

RkR$,,,t = gXk (klmn) (k'l'm'h), I h

I cause des relations (2.2) (en ce qui conceme les parentheses).

(iii) On peut ecrire les 96 composantea du tenseur $, explicitenlent en fondion m

de FA, et de leu- dBriv&s, ~avoir (1.3), d'oh dhoulent d m e n t les relations


(2.3), (2.4), (2.5) sont des propriet6s caract6ristiques du champ sym6trique, ex- prim6es par les composantes du tenseur de courbure R.

3. CAS 11. Quand le champ est diseym6trique, les conditions d'integrabilitR, dm equations (B) peuvent aussi se dBduire avec fiuxilite.

Avec (B) on considh aussi le systeme de leura d e r i v h premieres (B,); entre (B) et (B,) on BLirnine toutes les dBriv&s premieres de gfip et en ajoutant les conditions

on arrive immediatement aux conditions d'intdgrabilite suivantea :

d'ou l'on peut deduire immediatement celles-ci :

(i) En multipliant (3.1). par g,, et en' sommant ensuite (comme $'g,h = 4, grdg,, = $ et 8' sont les symboles de Kronecker), on a

Pm+S? ,=O I)

(ii) Comme m et p eont dea indices antisymBtriaues, (3.1) p u t ausai s'6orire comme ci-deasoue

en posant h = m et p = n, et apl* les contractions indiquks, on obtient la relation

Pour obtenir dw relations plus gBn&aJes, now oommenmns par un changement de notations pour Ies bomposentes des tensem R et 5.

Comme il y a seulement six paires de (m, n), c'est-&-dire d'indices antiaymmetriques, noue remplacons chaque (r) dans les composantes par un des nombres r = (r = 1, ..., 6) qui eat transfBr6 sous le symbole alphabetique R ou S du temur .

Suivent le nombre r les 96 Bquations s' arrangement maintenant en six groups oomportant chacun 16 relations.

Mais lee fA =fl, Rz, Sx,, etc., ae laissent interpreter comme les 8Bments de certaines matrices carrbs B, R et 8 (les indices indiquant maintenant lea lignes et les colonnes suivant des conventions bien connues), et par cosBquent les relations (3.4) peuvant itre repr&ntees commue une seule Bquation de matrices :

RB +.as' = 0 (3.5) r r

Certaines Cons~quences 1'Existence du Tenseur g

En prenant les fornlee t r a n s p o d , on a aussi

h s 6puations (3.5) et (3.6) jouiasent de propri6& d'hermitioitd, ainsi qu'il a Bte postul6 par Einstein pour toutes les relations de la th6oire unitaire.

En effet, on passe de (3.5) a (3 .6 ) en transposant G et en changeant le tenseur R en S et vice versa, ce qui implique que,les dites relations restent inalt6rbs cornme un tout par len changenients simultan6s des yP'' en gl'fi et des r;,, en I'$ : exactement conlme l'exige le principe d'hermiticitb.

Multipliant finalenlent les six Bquations (3 .6 ) par des nonlbrw arbitraries A,, et faisant la somme, on ++, apr6s une transformation facile :

G-'RG := -& (oh R = h,RI ...). (3.7)

Les nlatrices R et ~ s o n t donu liQs par une transformation qui laisse intaots leurs invariants, d'ou

l , ( R ) - ( - - 1)'lr(S) = 0 ( r = 1, 2, 3, 4). (3.8)

(hulne les A, sont tout a fait arbitraries, on p u t dmet t re que chaque ctoeffioient de yolynome en A, dens (3.8) pris e6par6ment eat Bgal it z6ro.

Dans toutea les relations qui s'8nsuivent entrent seulen~ent ri, et leurs dBrivks, chaoune d'elleb exprimant les propri6tCts caractdristiques du champ unitaue condition- ne6s par l'hypothhse de l'existence du tenseur de potentiel.

Si l'on adopb pour leu produits de la chahe des R ou des S les notations suivantes : r I

lea relations obtenuev ci-dessus purront s'6crire aussi cornme suit

R(r) = - #(r) ,

R(r, a) = B(rs) = S(sr) , 7

R(rat) = - s ( s t r ) = - S(rts) = -S(s1.1), i (3.10)

X(ratu) = S(r&r) = S(rub) = . . . J

4. (1. Cvmnle cons6auencea de I'hypothCne de I'existence du tenseur de potentiel, dans le champ des connexions aymm6triques, on a Ies relations su ivanh :

16 rsl&tions :

Compte h u dse 10 Bquthne du ohamp, on a en tout 77 reUiollcl hom@m entae lea cornpaeantea pm et leuns d&iv& pre~mih .

.W lea condit.iom qui mnt ~~ m s d;l6tuenk -aliinw, b t k k e unitaire, mlt bhn plus mtrictivw que celles ( p i w m t irnpwk aux 6lhenta du o h p de I* gmvitatiot~.

Reprinted ftom Lr Jow & Php et Je Radium (Paris) 14, pp. 646817,1@58.

Certain Consequences of the Existence of the Tensor g in the Affine Relativistic Field

By S. N. BOSE, University of Calcutta.

Smmmary - Relatiom are eetabliahed where only r k, and their derivativee enter from conditions of integrability of the equatione which the tenmr gof the relativietic theory eatiefiee.

Thenumber of independent relatione between the elements of the field ie much larger in the unitary theory than in the theory of gravitation.

1. Introduction The abstract formalism of the theory of relativity devdoped from the concept of the field of coefficients of affine co~ec t ion of a four dimensional variety

If W are the components of a parallel and infinitely small displacement to which a vector A of the chosen variety is subjected, the variations of the components A" which result are obtained by the rule

8A. = - AaQ W (1.1) In the theory of gravitation the terms B are symmetric, or IY, B = Q, ; thus the rule

of transport there is unique. On the contrary, as the coefficients are not symmetric in the unitary theory, one has here a second rule of transport :

Based on relation (1.1) we can easily calculate the tensor of curvature R, the components of which a = rL,. - r;,@ + TLC, - GC. I (1.3)

are antisymmetric in the indices p and v ; and then by contraction we obtain the Einstein teneor E with the components

But the unitary theory also admits (1.2) as a rule of transport. Thus the same reasoning mentioned above will give us from (1.2) hother transport S (replacing R) with its components


and finally also the tensor H (replacing E of Einstein) with the components

The equations E, ,, , ,, are those of the field in the theory of gravitation. The ambiguity of the law of transport in the unitary theory seems to make it difficult

to choose E,, = 0 as fundamental equations. But the difficulty has been solved by Einstein in the following manner : it imposes four restrictive conditions on the affine field, namely

- r;p = 2r,, = 0. (1.7)

Though arbitrary, this hypothesis easily gives

the two laws thus lead to the same tensor. Therefore the field equations can remain the same in both the theories.

Besides the system

which depends only on the coefficients cv and their first derivatives, there is also a second system where the components of the potentialgp " and their derivatives are also included ; these are indeed the equations

We have demonstrated that (A) and (B) are obtained from one variational principle. We form the invariant integral

with some components of the potential and the Einstein bnsor (supposed ta be unique due to rp = 0) and we hold

61 = 0

for all arbitrary variations of g'" and cv ; we immediately obtain the required equations.

It is evident that the existence of a potential tensor in the affane field would give it a very special character.

Indeed (B), regarded as a system of differential eequations, can determine gpv only when the conditions of integratibility are satisfied.

Hence the relations, which follow as consequences of this hypothesis of integrability between the coefficients I$, and their derivatives will express those relations of the

Certain Consequences of the Existence of the Tensor g 303

characteristic properties of the field which are implicitly postulated by the theory of relativity as well as the theory of the unitary field itself.

To obtain them in an explicit form is precisely the aim of this work.

2. Case I. -When the coefficients riV are symmetric, the conditions of integrability of the equations (B), namely :

can more easily be expressed in terms of the curvature tensor R. If the components R :, are expressed by the multiples gpv and brackets with four

n

indices as EII, = g" ( k h n ) ,

n (2.1)

the conditions of integrability will be well expressed by the following properties ofthose brackets which are well known :

(klmn) = -(lkmn) = (mnkl) = -(nmkl). (2.2)

From these we simultaneously eliminate gpv and the brackets and obtain relations between the components of R only.

R h = g" (klmn), n

we hold h = I and contract. We now observe that the brackets are antisymmetrical in the indices (k and I ) while g Ik are symmetrical.

We then automatically have :

Rim = 0

(ii) Similarly, since n

R&,R;4,, = g X k (klmn) gnk' (k'llm'A), n X

R s Riqt = $C'" (nmlk) gdX ( A m'l'k'), we have k E'

due to the relations (2.2) (with regard t o the brackets). (iii) We can write the 96 components of the tensor R explicitly as functions of

m

r \,, and their derivatives, that is t o say (1.3), which easily gives the relations


(2.3), (2.41, (2.5) are characteristic properties of the symmetrric field, expressed by the components of the curvature tensor R.

3. Case 11. - m e n the field is non-symmetric, the conditions of integrability of the equations (B) can also be easily deduced.

With (B) we also consider the system of their first derivatives (B1). We eliminate all the first derivatives of gv between (B) and (B1) and using the conditions

we immediately come to the following conditions of integrability :

from which we can immediately deduce these : (i) By multiplying (3.1) with g, , and then summing (since , = 6;, g p g , , = $

and 6 ' s are the symbols of Kronecker) we get

(ii) As m and p are the antisymmetric indices, (3.1) can also be written as

by putting A = m and p = n, and then after the contraction indicated, we obtain the relation

In order to obtain more general relations, we begin with a change of notations for the components of the tensors R and S.

As there are only six pairs of (m, n), i.e. antisymmetric indices, each

R or S of the tensor. component is replaced by one of the numbers r ( r=l , ... , 6) which replaces

Thus (3.1) take the following form =

Following the numbers r, the 96 equations arrange themselves into six groups each having 16 relations.

Certain Consequences of the Existence of the Tensor g 305

But ga ' = g: , R: , s;, etc can be interpretated as the elements of certain square matrices G, R and S (the indices now indicate the rows and columns as per the well known conventions), and consequently the relations (3.4) can be represented as a single equation of the matrices :

--

where S represents the transposed matrix of S with - s; = Sf.

By considering the transposed forms, we also have

The equations (3.5) and (3.6) have the properties of hermiticity as i t was postulated by Einstein for all the relations of the unitary theory.

In fact, we pass from (3.5) t o (3.6) by transposing G and changing the tensor R and S and vice versa, which implies that the said relations remain unaltered as a whole by the simultaneous changes ofg" into gV ' and of T", into Tt ,, exactly as per the principle of hermiticity.

Finally, by multiplying the six equations (3.6) by arbitrary numbers h . and summing up, we get after an easy transformation :

The matrices R and are therefore linked by a transformation which leaves their invariants intact, from which

As the h , are completely arbitrary, i t can be said that each coefficient of the polynomial in h , in (3.8), taken separately, is equal to zero.

In all the relations that arise only Tiv and their derivatives enter, each of them expressing the characteristic properties of the unitary field conditioned by the hypothesis of the existence of the potential tensor.

If the products of the chain of the R or the S are given the following notations r r r

the relations obtained above can also be written as


4.a. As consequences of the hypothesis of the existence of the potential tensor in the field of symmetric connections, the following relations are obtained :

16 relations :

6 relations :

(ii)

45 relations :

Taking into account the 10 field equations, we have a total of 77 homogenous relations between the components ri, and their first derivatives.

b. In the unitary field, we have :

6 relations :

21 relations : R(rs) = S(sr) ;

56 relations : R(rst) = - S(tsr);

126 relations : R(1Ju) = S(utsr).

Coupled with the field equations the total can go upto 225 relations between the same elements of the field.

Thus the conditions which are imposed on the affine elements in the unitary theory are more restrictive than those that are imposed on the elements of the gravitational field.

Manuscript received 18 July 1953.

English translation of Le Jour de Phys et le Radium (Paris) 14, pp. 645-647, 1953.

THE AFFINE CONNECTION IN EINSTEIN'S NEW UNITARY FIELD THEORY

(Received September 29, 1952)

The non-symmetric I'4,'s are defined by the 64 equations

in the new theory. A method of obtaining explicit expressions of I"s in terms of g,, and its derivatives is sketched in this note. Following the usual convention of the relativity theory, the summation symbols are suppressed throughout, i t being understood that summation is always implied whenever dummy indices occur.

I. The generalized Christoffel bracket (:') is defined by the equation

(the subscripts are written in a cyclic order (Schrddinger)). Also,

where s, a, denote symmetric and anti-symmetric components of g. If r", is similarly decomposed into

and the bracket (Z") into

G'l = +[@') + (:")I (5) and

{ Z V } = + [(ZV) - (:,)I the following relations follow immediately


Again if

we can espress S and T' in t.erms of P and T with the help of s", the contravariant component, of s,, obtained in the usual manner; also taking

as the components of the matrix product of s and a, we can rewrite the equations in the form

P's can now be easily eliminated, and a tensor equation, involving only T's is obtained in the form

This tensor equation may be regarded as characterizing the ~sntinuum of the Unitary Theory.

When (B) is solved, yielding T in terms of known functions of g and its derivatives, and also &s, t,he substitution in (A) of T immediately gives the P's and hence also the r*s.

To solve this central equation we first observe that a cyclic addition easily yields

which is easily recognized as a tensor. at pv Also if Christoffel co-efficients r:, = s [ r ] (the usual symmetric 7's) are used,

Connections in Einstein's New Theory

the lefbhand side of (B) is changed to

where

is the familiar covariant derivative with symmetric y",'s as in Riemann-Geom- etry ; also observing that

the equation (B) can be transformed into the form

where the anti-symmetric suffixes are put in the end on both sides in the tensors with three-index symbols.

11. To solve the equation (C) we remark that the co-efficients C that occur in the equation are the components of the matrix

C = sa = 11 C i (1 o r C i = s k t a t p .

The general equation for the eigen vector M of the matrix C can be written in the form

(10) ( C - p E 1 M = 0 or C ~ , M ; = p ~ i ;

in the vector M = M ," the superscript a indicates the component, and the subscript p the associated eigen-value.

The characteristic eigen values are the roots of the quartic

as the first and third invariants vanish (the matrix being the product of a symmetric matrix).

A few well-known properties of matrix equations are here re- capitu- lated for ready reference.

l.A matrix C, and its transposition C have the same eigen values, but different eigen vectors ; and the eigen vectors, written with two suffixes, (representing the component and the eigen value) satisfy the following relations :


1 (a). The theorem is true even in the case of repeated roots of the characteristic equation of C, if the eigen vectors are suitably constructed.

2.Taking the elements of the matrix C, as C k,, where k denotes the rows and p the columns of the matrix, a new compound matrix C x C , can be constructed in the following manner :

obtained by the usual cross-multiplication rule. The elements can be arranged in a six x six matrix in any proper order ; e.g.,

3.The eigen vectors of the matrixes C x C (arranged in any assumed manner) may be obtained by taking the vector product of the eigen vectors M i of C. Ma x Mb is thus the eigen six-vector of the six matrix, C x C corresponding to the eigen value papb.

4.The eigen vector of C x C is similarly obtained as Ba x Hb corresponding to the same eigen value p g b .

5.The eigen vectors of the compound matrices have corresponding orthogonal properties.

The components of the eigen vectors can be written by the notation used before as f i f ($ ) and a ( $ ) .

The orthogonality relation is written as

111. Keeping the above well-known properties of matrices in mind me can tackle the problem of solving the general equation. We multiply the equation (C') by M:MfJfE formed from the eigen vectors of C, corresponding to three eigen values a, b, and c, where b # c; then remembering the equations

c$W~ = pbhl: , Ctitl: = p,.ll: , etc., (b, c pot summed)

we get easily, by suitably changing the dummies

1 b T = f h k l ] ~ ~ n f ~ H ~ ( 1 + pbpc) + a d & X . M : ;

(14) I also

Connections in Einstein's New Theory 311

Or remembering the anti-symmetric nature of the suffixes li, 1 we write the same equation with I;, 1 in the reversed order and add, using the symbol

easily getting the result in the form

Remembering the orthogonal properties of the eigen vectors, namely equations (12) and (13), and multiplying by @%(Z;) on both sides followed by summation on a, b, c, we have

An explicit form of Tap, is thus obtained in terms of the eigen vectors of C and 6 and the covariant derivatives of a k l calculated with the symmetric Christoffel co-efficients, 7's obtained from the symmetric parts of g,, , namely s,, . The discriminant-determinant of the equation (C) is obviously

(17) D24 = n (1 + papb + papc + p,p.)(product with roots of (11) with b # e).

The value of the determinant can be easily calculated in terms of the invariants of C; in fact,

The determinant D always vanish when 1 + Iz + 14 = 0; but this happens when the determinant of g = 11 g,, 11 = 0.

This is proved as follows: Any transformation s( )$-I, where s-I = S, preserves the symmetric or/and

the anti-symmetric nature of a matrix; s,, part of g,, may be diagonalized, and at the same time the anti-symmetric tensor a,, preserves its anti-symmetric character. Hence g can be reduced to the form

sl1 a12 . . and a,, + a,,, = 0


by such a transformation. The values of all invariants remain obviously unaltered. In this special co-

ordinates

(19) A = I/ g,. 11 = ~ 1 1 ~ 2 2 5 3 ~ 4 4 + 2511s22& + (alza~ + .I.* The invariants of the matrix C can.be easily calculated which gives

That when T is determined I"s are also determined follow simply as a consequence of equation (A). IV. In later developments of his theory Einstein has introduced the additional

condition

however, from the contravariant form of equation (I), viz. :

the following relation

meam four relations involving the differential co-efficients of a,, and g,, . In the equation ( C ) for the determination of T, all first order differential co-efficients occur in the left side of the in-homogeneous equation, whereas the homogeneous

Connections in Einstein's New Theory 313

part involves only C; , composed of s" and a,. and no differential co-efficients occur here. Hence the additional conditions do not affect the general nature of the solution.

Reprinted from Ann Math USA 69, pp. 171-176, 1964 (Princeton University, NJ).

XVIII. A REPORT ON THE STUDY OF THERMOLUMINESCENCE

By PROFESSOR S. N. BOSE, J. SHARMA and B. C. DUTTA, Khaira Laboratory, University College of Science, Calcutta

(Received for publication on February 9, 1955)

When some solids are irradiated with ionizing radiations like X-ray or cathode rays, they often exhibit fluorescence followed by a long period phosphorescent afterglow. Again, if the irradiated sample is heated, the stored energy is released in the form of radiation, emitted at various temperatures characteristic of the sample. This thermo-stimulated release of energy is commonly known as therrnoluminescence. The capacity of substances to store energy is large at low temperatures, so thermoluminescence is pronounced if the irradiation is carried out at low temperature (e.g., at -184OC). We may plot in a thermoluminescence curve the total intensity obtained from the sample at various temperatures during heating. From the curve, the trap depths of light-storing states below the conduction band, can be evaluated 1. Shallow traps yield glow peaks at comparatively low temperatures, while deeper traps are released at higher temperatures. Quartz, alkali halides, calcite, glasses and many other substances show this kind of thermoluminescence. Of late, thermoluminescence has been a useful tool in many problems of research. Dosimetry of radiations, study of heterogeneous catalysts, ionic nature of elements in glasses, and identification of clay minerals are. some of the uses to which therrnoluminescence has been put.

Study of therrnoluminescence of alkali halides was initiated in this laboratory in 1950, as a part of a larger programme of investigating solid state and of co-ordinating fluorescence study with the results of soft X-ray spectroscopy. The fluorescence spectra of alkali halides both at room and low temperatures had already been studied 2*a 4. Measure- ments of the afterglow 5 decay rates had also been done, so therrnoluminescence study of alkali halides and also of some organic substance was undertaken, with a view to clarify the processes involved during energy storage in these phosphors.

In this laboratory, therrnoluminescence has been studied by exciting the sample at liquid oxygen temperature, in a demountable cathode-ray tube fitted with a window of quartz. The sample is mounted as a fine coating on a hollow bulb made of thin silver. The bulb can be filled with liquid oxygen to keep the sample at low temperature and it may be rapidly heated with an electric heater immersed in the bath. A thermocouple on the surface of the bulb, connected to a Moll galvanometer, gives temperature of the sample. Another galvanometer connected to the photomultplier tube, placed facing the specimen, gives luminescence intensity. The movements of the two galvanometers during thermoluminescence are recorded on a slow rotating photographic drum. A typical example of thermoluminescence curve is shown in Fig. 1.

For recording therrnoluminescence, different photomultiplier tubes (931A, 1P28; 1P22) have been employed to suit the emissions of the different samples. Fluorescence spectra of the substance being known from a previous study, there was no difficulty in selecting the photomultipliers. Although low rate of heating (0.25"C/sec.) is usually preferred, yet it has been observed that, in the case of many phosphors, the weaker peaks get flattened out unless the rate of heating is high (6-10°C/sec.). A high rate of heating

I2

A Report on the Study of Thermoluminescence

FIG. 1. Thermoluminescence of KI.

was often maintained and a fine film of powdered sample was found more suitable than a single crystal. For recording the spectral nature of the different glow peaks, two photomultipliers, sensitive in different regions, were made to record thermoluminescence simultaneously during the same run of the experiment, and optical filters were used to separate out the emissions.

Quite a number of alkali halides, both pure and activated, has been studied and some of the results have been publishedo> 7. These halides yield a number of thermoluminescence peaks between 90°K and 600°K. It has been possible to explain some of the results of afterglow decay rates in the light of information obtained from thermoluminescence study. The effects of thallium activation of samples on thermoluminescence have been investigated and it has been found that in some samples thallium produces a new peak of its own, without affecting the peaks of the parent lattice. But in some samples, like KC1 and KI, it completely changes the thermoluminescence curve. From the comparative method of study with optical filters, it has been found that the different peaks emitted by a specimen are not spectrally identical. In general the peaks emitted at high temperatures are rich in emissions of ultra-violet and blue regions. But thorough study of any peak with an ordinary spectrograph is not possible, because the individual peaks disappear within a few seconds. To study the spectral nature of individual peaks, construction of a rapid spectrophotometer was deemed necessary and has been recently completed. With the help of this spectrophotometer some interesting results have been recently ~btained (see Figs. 2, 3, 4 and 5).

The automatic rapid scanning spectrophotometer employs two concave mirrors of 25 cm. focal length and large focal ratios offl4.2 constructed from stainless steel and a


large prism of quartz. Light from the first slit is collimated by the fist concave' mirror, and, after dispersion by the prism, the beam is focussed by means of the other concave

minor. But before falling on the exit slit, the spectrum is reflected by another plane minor. The latter is mounted on.-* turnteble, which can be made to oscillate to'and fro with the help of an induction motor and a cam. The entire spectrum (ultra-violet and visible) sweeps before the slit in about 0.95 second and flies back instantaneously to the original position by a cam arrangement. Just behind the exit slit is placed the photomultiplier tube which receives the light. The output of the photomultiplier tube passing through a pre-amplifier is amplified and displayed on a Dumont 304H cathode-ray tube with long time. A micro-switch attached to the cam triggers 'the sweep circuit every time the ultra-violet end of the spectrum just comes near the slit so that the same part of the spectrum is given at the same position. During each sweep, a specially adopted syn- chronous 35 mm. camera arrangement photographs the intensity distribution curve displayed on the cathode-ray screen. With the help of this spectrophotometer, the spectral intensity distribution of the emission can be photographed in less than one second and the apparatus is capable of giving all the necessary information during the entire thermoluminescence experiment. Not only the spectral composition and the intensity of the different peaks can be obtained, but also the changes in emissions in subsequent periods of a single thermoluminescence can be easily and accurately studied. Commercial types of photomultipliers were found to 6e not sensitive enough for this work, so a special photomultiplier with 19 dynodes was procured from France.

It is well-known that most of the solids (like alkali halides, quartz, calcite), on irradiation with ionizing radiations, develop colour centres. From simultaneous changes in colour centres and by recording diffuse reflectivity during thermoluminescence, it has been shown that some of the luminescence peaks correspond with.thermally actuated transformation in colour centres. It has been found that thermoluminescence-behaviour of alklali halides is very susceptible to previous history of heat treatment of the sample. Other properties have also been studied in this laboratory.

Thermoluminescence of some aromatic hydrocarbons both solids and liquids between 90°K and 250°K has also been investigated. This work gives information regarding the metastable states of the molecules excited by cathode-rays at low temperatures. The depth of the metastable states below the upper excited states, as measured from the thermoluminescence, agrees fairly with that obtained from the study of fluorescence and phosphorescence spectra. Thermoluminescence opens a new method of investigation of the metastable states of the molecules.

This article would not be complete unless another interesting application of thermoluminescence is mentioned. We know that many naturally occurring minerals exhibit thermoluminescence. Recently a number of samples of sands procured from various parts of this country was studied. All sands show feeble thermoluminescence, but those from Bargarh and Mangalhat showed pronounced one. Measurement of Y-ray counts in the case of the latter showed slightly extra counts oyer the background. This leads to the possibility that thermoluminescence might be developed as a preliminary method of detecting radioactive minerals. Many minerals (like bentonite) are often found to yield thermoluminescent radiation. The ultimate reason for the occurrence of thermoluminescence in natural minerals has not yet been found. Radioactivity may play some part in this occurrence, but it is yet too early to make any definite statement on this matter.

A Report on the Study of Therrnolurninescence

1. Randall, J. T. and Wilkins, M. H. F. (1915). Proc. Roy. Soc., A 184,365. 2. Bose, H. N. and Sharma, J . (1950). Proc. Nat. Insr. Sci. India 26, 47. 3. Chatterjee, A. (1950). Indian J. Phys., 21, 331, 266. 4. Bose, H. N., Sharma, J . and Chatterjee, A. (1952). Proc. Nut. Inst. Sci. India, 18, 389 5. Bose, H. N. and Sharma, J . (1953). Proc. Phys. Soc., (London), B 66,371. 6. Sharma, J . (1952). Phys. Rev., 85, 612. 7. Bose, H. N. (1955). Proc. Phys. Soc., London, under publication.

318

Bosc, Sharmtr arid Dutta


Trans. Bow Inst., Corn. Vol., Plate I.

FIG. 2. Shows the fluorescent spectra ol'anthracene.

PIC;. 4. FIG. 5. FIGS. 3, 4 and 5 give the radiation as observed from a sample of irradiated NaCl during heating.

(Note change of character with time.) Figs. 3, 4 and 5 are photographs taken at 3, 7 and 17 secs. respectively after the commencement of heating.

Reprinted from ! h n o Bose Ru Znst 10, pp. 177-180, 1955 (Bose Institute, Calcutta).

SOLUTION D'UNE EQUATION TENSORIELLE INTERVENANT DANS LA THEORIE DU CHAMP UNITAIRE;

PAR S. N. BOSE,

Universite de Calcutta.

I<Bsume.--On exprime explicitemeut en fonction dcn C et de Ileum invariants la solution de 1'6quetion

Lensoridle d'oh depcnd la det,erminat,ion des P X Y "'

On remarque que l'itkr~tiotl fournit une solution rigo~~reuse quend det C = o

1. Dans un precedent article "), on a montr6 qu'il est possible de ramener le calcul des coefficients rtffines pk, en fonction des g,,, et de leurs derides & la solution de 1'6quation tensorielle

Les tenseurs U et T sont antisymetriques par rapport aux indices /L et v, Cf sont 10s Blements d'une matrice C form& & partir des parties symm6triques e t antisym6- triques du tenseur q :

Les v~ leurs propers z de 1 matrice C vkrifient 1'6quation

(1.3) x4+I ,xa+I , = 0>

1, et I 4 sont les invariants d'ordre pair de C; de 1'6qustion (1.2) il r6sulte imm6diate- ment que les invariants impairs sont nuls.

Le vecteur propre M t de C correspondant Q la valeur propre a vdrifie 1'6quation

(1.4) C$M; = aM;.

La matrice transposk C(G = Ci) a lea m h e s valeurs propres que C mais ses

vecteurs propres 2: sont differents.

Si l'kquation caracteristique a toutes ses racines distinctes, il existe une s6rie

complbte de vecteurs propres M et M de C et 6 et l'on peut exprimer lesBl6ments de C en fonction des vecteurs propres : Savoir

On calcule facilement la solution de (1.1) :


= Z ( l + a b + b + w ) - l ~ : ~ ; ~ ( ~ ) k i ( ~ ) ~ , (b # c), T". &

(1.6)

et la sommation porte sur lea 24 oombinaisona possibles des (abc).

On p u t Ccrire \a solution (1.6) sous la forme d'une Bquation tensoriellr! T = BU,

le tenseur B nyant pour composantes :

(1.7) est identique a (1,5).

J e me propose de faire la son~mation dans.(l.'i') e t d'obtenir les composantes de B en fonction des Bl6ments de C.

2. Au moyen de C et de la matrice unitaire E nous pouvons former une algkbre {C,) avec addition e t multiplication matricielle.

La formule g6n6rnle d'une mntrice A dans {C+) est

i A = ~ o + l ~ l C , + ~ 2 C 2 + ~ 3 ~ 3 ; C, = C, C2 = CC, C, = CCC.

Toutes -1es matrices A commutent et ont donc les, m6mes vecteurs propres M que C.

- Les matrices 2 de IYalg6bre trnnsposBe ont a u ~ s i les meme8 vecteurs propres M

que C.

Si a, ;, b, b sont les racines de 176qtmtion c;n.actCristique (1.3), on voit f~cilement qne les matrices de base se tlBveloppnt ninsi :

oh f (a) = Z,M,. . . . en njoutant lea indices infhrieurs . et sn$rieurs AUX formules ci-dessus. onobtient

D6dui~ons inahtenant line nouvelle nlghbre {V,) : oh l'on fortne a i n ~ i les matrices V B prtrtir de deux 616ments quelconques A, B, de [C,) :

V = (AB) ,

Solution d'une Bquation Tensorielle

avec

La rhgle de composition (2.3) montre que toutes les matrices V de (V,) con~mutent entre elles.

Toutes les matrices V ont les mGnies vecteurs propres h. six composantes que (CC); on voit facilement que les six valeurs propres de (CC) s'obtiennent en multipliant deux h deux les valeurs propres de C. Ce sont

(2.4) a6, bb, ab, ti&, a&, tib;

e t les vecteurs propres it six composantes se deduisent des vecteurs propres de C par une rbgle analogue. Q (2.3) par exemple :

(2.5) M,"; = M i IZIX-IM~M;,

sont cles coinp~sicntes de iM (ub) le vecteur qui correspond a lu valeur propre ub de (CC).

Ties n~i~trices v de I'ltlghbre transposee {v,) ont icussi les mihnles vec te i~~s propres

A six co1npos;~ntes yue @): on voit fwileinrnt yue les vecteurc; ~ ( a b ) se tl6duisent dea

vecteurs.propres de C par la rhgle xavoir. - .- - - M i ; == M ; M;j-M$ M;, . . .

Comnle bases de l'alghbre {V,',, nous pouvons prendre six matrices quelconques de T7, linkairement indhpendantes.

Nous choisissons les six matrices suivantes dont nous donnons le &I-eloppement en fonction des vecteurs propres :

Let formules (2.6) sont obtenues d'une manihre t,out Q fait analogue a la niitnihe dont on a obtenu les formules (2.2).

Les quatre autres combinaisons yui peuvent ixtrt: formBes A pi~rtir des elements

(2.6) 4

( E E ) = V(aC + ~ ( b b ) t V(ab)+ ~ ( d b ) + Ir(a6) + V ( E b ) ,

(CC) = -a2 ~ ( u 6 ) - b2 v(bb)+ab[v(ub) -I- V(db)l -ub[V(ah)-+ T7(Eh)]

(G2C2) = U ~ T ~ ( U G ) -+- b4v(b6) +u%'I V(ah) -r i7(CD) ] -t (~'6" L'(ab) V(Gb)J,

2(EC2) = - i !a2~(nG) -2b2v (bb )+ (a2+h2) [ V(uh+ v (dh )+ 17(ah)+ V(Ab)],

2(EC)= -(a+b)[V(ab)- v ( ~ b ) ] + ( a - b)[ V(ab)- V(Gb)],

2(CC,) = ab(a+b)[V(ab) - V(Eb)]-ab(a-b)[v(ab) -- V(Eb)].

. V(ab) = %(ab) M(ab), ...

322 S N Base : The Man and His Work

de base de (Cg) dependent des six ci-rlessus ell vcrtu des quatre identit& ci-dessous :

I (EC,)+(C C,)+I,(EC) = 0 , !2(CC,) +(C,C,) + I,(CC) - I ,(E:#) = 0,

(2.7) (C2C3) --I,(EC) = 0, (C3C3)-Iz(C2G2) - 14[(CC) -1- 2(EC2)J = 0.

On ~no~itre aussi facilement qn'en ~nultipliirnt Vl et V , dnns [ V,,) w e c

(2.8): V1 = ( A B ) , V2 = (CD), on a

VIV, = #:(AC . BL))+(ADBC)J.

Avec (2.8), leu identites (2.7) et I'identit6 generale de Cnyley-Hamilton :

(2.9) C,+I,C,+I,E = 0

nous pouvons r6duire toute matrice de forme (C,C,) B une fonction lin6aire des six 616ments de I s base t~dopt6e en (2.6).

On p u t choisir des bases et dBmontrer des identities analogues pour toute lzlgbbrr V,,,-l, - composQ B yartir de {C,) [par la manibre analogue & (2.3)].

2

.Arec les mineurs m3 d'une matrice C, quelaonque, nous pouvons former la matrice conjugu& D.

Dens le cas particulier de C ici Btudih, oh C =5 Sa est le produit de composantes eym6triques S par les antisym6triques a, on voit facilement que D est donne par

D = A'd,

oh A' et 8' sont form& de meme B p&ir des mineurs m8 de a et 5.

Admetton que toujours

On p u t cependant avoir

Le oas I( a ( 1 = 0 est inthressant pour la thbrie du champ unitaire. Nous dis- outerons donc lea propri6t6s particulikresde V6 quand 1 1 C 11 = 0 par suite de (1 a 1 1 = 0 .

a eat une matrice antisymhtr'ique d'ordere 4. Donc A d6terminant de a ast un oarr6 parfait et 2/~-est un fa~teur dr? tous les mineurs rn, de a,.

W 1 ) a 1 1 = 0, A et D mineurs conjugu6s de C , sont identiquement nuls.

Your la matrice oonjugu&, nous avons une identit6 analogue B (2.9) ; ( a . l o ) C,+I,C,+ D = 0 (C, = CCC).

Quad

(2 .11 ) (1 a ( 1 = 0 , I , = 0 et D = 0,

Solution d'une &quation Tensorielle

et des identittis (2.7) nous tirons imm6diatement

(2 .12) (CC,) = 0, (C,C,) - W ' l C , ) , (C,C,) = 1,(C2C2).

Nous allons utiliser ces relations pour dkduire la solution de ( 1 . 1 ) dam le oas particuIier I, = 0.

3. Nous effectuons mitintenant la sommation indiquQ en (1 .6) e t (1.7). Notons quelques propriktk simples du denominattiur D ( a h ) :

A l'aide de ceg relations simples, now pouvons regrouper faoilement les termes 8. eommer :

On utilise les notations ci-aprks pour Qcrire le resultat final :

( V(ab)+ V(&) = X , v(u&)+ V(6b) = Y ;

(3 .2 ) V(ab)- V(d6) = U , ~ ( a & ) - V(&) = V ; B , M , = f(a) , @ ( b c ) ~ ( b c ) , = V(bc), .....

La sommation (1 .7 ) est notde

Aprhs regroupement, il vient

S N Bose : The Man a d His Work

Servons-nous de ( 2 . 2 ) et ( 2 . 6 ) pour exprimer X , Y, U, V et f ( a ) f f ( 6 ) , ... ; X , Y, U, V en fonction de matrices de ( V J et {C$, et nous obtenons ap rb un calcul simple le r6sultat sous sa forme definitive

oh les coefficients d, g, p, v, f, h sont fonctions des seuls invariants de C. Si

sont fonctions des invariants.

Les coefficients de (3 .5) sont

4. Nous avons exprim6 le r6sultat find (3.6) en fonction de matrices d6duiks de C mivant des lois deterrninhs, avecdes coefficients qui sont fonctions des invariants de C. Ce resultat est valuable dans tous les cas, mbme quand pour des matrices parti-

Solution d'une gquation Tensorielle 325

culieres, il n'existe pas un systame complet de vecteurs propres et l'on ne p u t plus developper suivant (3.3) la; matrice rBsolvante B mais le resultat gBndral (3.5) reste toujours vrai. L'expression (3.6) donne donc la solution gBnBra1e du problame pose en (1.1).

NBanmoins, il est interessant d'observer que, dans les ow particdiere I4 = 0 ou I , = 0, I , = 0, on peut fachnent mettre la solution sous forme d'une somme d'un petit nombre de termes, p i c e B, une method6 d'iteration.

Discutons brihvement le cas I , = 0.

On a d6j d6duit les propriBMs specides de la matrice .C en (2.11) et (2.12) : h sav0ir

( ' 3 7 2 ) = 0, (CaC,) = ~,(C,CJ, (C,C,) = ~,(C,C,)'

C3+12Cl = 0, C,+I,C, = 0.

Nous Borivons (1.1) dans la forme

U = AT = [E{(EE) + (CC))+SC(EC)]T

ou

(4.1). U = T+[E(CC)+2C(EC)] T = T+ST;

et cherohons une solution en posant

T = U+@,.

L'Bquation ayant la forme

U = T+8T.

On deduit fadement pour a, l'equation

-8U = Qb1+8@,-

En repetant l'ophtion, nous obtenons successivement

(4.2) 86U = @,+8@,, -888U = @,+8Q3.

En raison des proprihtkk sp4oiales de lYop6rateur [(2. I 1 ) -(2. I 2) ] nous ddduisons facilment

(4.3) 688 = I;&

En posant 8 3 A-E, oh E est l'ophrateur-unite, 1'Bquation (4.2) s'6orit

ou (4.4) A(Q3+IgU) = I iU.

L'Bquation originale eat AT = U.


Comme dBt A # 0, la solution est unique et l'on voit facilement que la solution de (4.4) est

Q3+4U = 4 T . En Bcrivant

cPS = T-U+GU-BGU = a ( T - U),

nous avons facilement

En d6veloppant, nous svons le resultat suivant :

La solution gBnerale nous donne le &me dsultat en posant I, = 0 et en utilisant les relations particulibres (2.1 1) et (2.12).

On p u t aussi traiter de mgme lea autres 1% = 0, I, = 0.

Si les coefficients h de N r Ao+AICl+AtC,+ h3Cs (nombre quelconque dans [C],)

sont ohoisis dans l'alghbre [V],, cee nombres forment une nouvele alghbre [cVla4.

egt donc Bquivalent B trouber l'hverse de. B en tant que nombre algebrique. L'inverse est aussi un nomber dam [cV] et on peut le trouver facilement de la facon suivante.

Nous vkrifions d'abord les r&ultats suivants t i partir de la rhgle de multiplioation et des identites pour l'alghbre [V],.

R = 2L+I,M, X = 4(EC)% = 2(CC)+2(Ec,), et D = Ig-414, le discriminant des Bquations

x%+Ifi+I, = 0, on a alors

L(EC) = M(EC) = 0; d'oh LX = MX = RX = 0

Solution d'une gquation Tensorielle

Dam ce qui suit, nous employons les abrdviations suivltntes : s = 2f I,, 2 , = 1+12+1,, z = 1+5I,-I,, et y = 2-1,.

Comme B = (EE)+(CC)+SC(EC), en le multipliant par son conjug6 B* = (EE)+(CC)-2C(EC),

nous avons BB* = (EE+CC)2--4C2(EC)2

BB* = (1+

aprbs factorisation;

Comme

1 [li-q;-M] [to+X(l+ 12)+C2X] BB* = [ to+X-C2X][to+X(1+12)+C2X]

aprbs 17addition du mcme facteur dans le numkrateur et le ddnominateur de I'expression de droite.

En faisant la multiplication, on obtient pour le ddnominateur

to[x+y(X+I2)+RI. Nous observons que

[x+Y(X+I~)+RI[X-Y(X+I~)-VRI = Q = t+t- = 22-4I4y2

On p u t donc exprimer faoilement 17inverse par 176quation suivante :

En supprimant les crochets, on obtient un dsultat 6quivalent B oelui de l'article principal.

Reprinted from Bull Soc Mgth France 83, pp. 81-88, 1956 (Gauthier - Villars Publishers, Paris).

Solution of a Tensor Equation Occurring in the Unitary Field Theory

By S. N. Bose, University of Calcutta.

Summary - The solution to the tensor equation on which the determination of c, depends, is explicitly expressed in terms of the C and their invariants.

Note that the iteration gives a rigoroua solution when det C = 0.

1. In a preceding article (I) it has been shown that i t is possible to reduce the calculation of the &ne coefficients c, in terms of g"" and their derivatives to the solution of the tensor equation

The tensors U and T are antisyminetric with regard to the indices p and v, C are the elements of a matrix C formed from the symmetric and antisymmetric parts of the tensor g :

1 1 c = " 3 = 5 ( g p r - g r u ) i 4 r = 3 ( g t k f g k : ) ,

(1.2) I spfstk = d[.

The eigenvalues x of the matrix C satisfy the equation

IP and I4 are even invariants of C ; from the equation (1.2) i t immediately follows that the odd invariants are zero.

The eigenvector M of C corresponding to the eigenvalue a satisfies the equation

The transposed matrix C ( C = C k, ) has the same eigenvalues as C but its eigenvectors El are different.

If the characteristic equation has all its roots distinct, then there exists a complete series of eigenvectors M and %3 of C and C and we can express the elements of C in terms of the eigenvectors : that is

Solution of a Tensor Equation

The solution is easily calculated from (1.1) :

and the summation is over the 24 possible combinations of (abc). The solution (1.6) can be written in the form of a tensor equation

the components of the tensor B being

(1.7) is identical to (1.5). I intend to carry out the summation in (1.7) and obtain the components of B in terms

of the elements of C.

2. By means of C and of the unitary matrix E we can form an algebra (CJ with addition and matrix multiplication.

The general formula of a matrix A in {CJ is

A = &+/LIC,+P&,+P~C~; C, = C, C, = CC, C, = CCC.

All the matrices A commute and therefore have the same eigehvector M as C. The matrices A of the transposed algebra also have the same eigenvector M as 75. If a , a , b , 6 are the roots of the characteristic equation (1.31, one can easily see

that the basic matrices develop in the following way :

- where f (a) = Ma Ma,. . and by adding the lower and upper indices t o the above formulae, one gets

c ! ~ = M:AZ: - - B ~ M $ ) a+(. ! B P M ; - B ~ M ~ "- i b 1 Let us now deduce a new algebra {V6), where the matrix V is formed from any two

elements A, B of (C4) :

17 = ( A R ) . with


The rule of composition ( 2 . 3 ) shows that all the matrices V of {V6) commute among themselves.

All the matrices V have the same eigenvectors with six components as ( C C ) ; i t is easily seen that the six eigenvalues of ( C C ) are obtained by multiplying the eigenvalues of C two by two. These are

( 2 . 4 ) aZ, bb, ab, a, d, 6b; and the eigenvectors with six components are deduced from the eigenvectors of C through a similar rule t o (2 .3 ) ; for example :

are the componen-ts of Mab, the vector whichcorresponds to the eigenvalue ab of ( C C ) . The matrices V oftke transposed algebra {Vs), also have t h e ~ a m e eigenvectors with

six components as ( C_ C ) ; it is easily noticed that the vectors M (ab) are deduced from the eigenvectors of C by the following rule : - - - -

Z:: = M t M ~ - I M $ M;, ...

As the bases of the algebra m6), we can take any six linearly independent matrices of V6.

We choose the following six matrices whose expansion we give in terms of the eigenvectors :

C (L!!E) = V(uC + ~ ( b b ) -+ V(ab) + ~ ( d b ) + t7(ab) + V(d6),

( (CC) = - a 2 ~ ( u d ) - h ~ ~ ( b 6 ) + a b l ~ ' ( u b ) + ~ ( d b ) ] - u b [ v ( a h ) j- V ( d h ) ]

I V(ab) = B(ab) M(ab). ... The formulae ( 2 . 6 ) are obtained in a manner completely similar to the manner used

to obtain the formulas (2 .2 ) . The four other combinations which can be formed from the basic elements of {C4)

depend on the six above mentioned equations by virtue of the four identities mentioned below :

(EC,)+(G C?)+I,(EC) = 0, 2(CC3) + (C2C2) + I,(CC) - I ,(E E ) = 0,

( 2 . 7 ) (C2C3) -I , (EC) = 0, (C3C3)- 12(C2C2)- [,[(CC)-I- 2(EC2)] = (1.

I t is also easily shown that by multiplying V1 and V2 in (V6) with


we have V1 == (AB) , V , = (CD) ,

With (2.8), the identities in (2.7) and the general identity of Cayley - Hamilton :

each matrix of the form (C, C,) can be reduced t o a linear function of the six elements of the basis adopted in (2.6).

We can choose the bases and demonstrate the analogous identities for all algebras Vn c n - 1 ) composed from {CJ [by a process similar t o (2.3)l.

2

With the minors m3 of any matrix C4, a conjugate matrix D can be formed. In the case of C studied here, where C = Sa is the product of the symmetric

component S by the antisymmetric a , it is easily seen that D is given by

where A' and s' are formed similarly from the minors m3 of a and S. We take for granted that always

One can nevertheless have

The case 11 a 11 = 0 is interesting for the unitary field theory. Therefore we will discuss the particular properties of V6 when I 1 C 11 = 0 as a consequence of 11 a 11 = 0.

a is an antisymmetric matrix of order 4. Therefore A which is the determinant of a is a perfect sequare and 6 is a factor of all the minors m3 of a.

If 11 a 11 = 0, A and D, conjugate minors of C, are identically zero. For the conjugate matrix, we have an identity analogous t o (2.9) ;

(2.10) C,+IaC1+D = O (C, = CCC).

When

we have

and from the identities (2.7) we immediately obtain


We will use these relations to deduce the solution of (1.1) in the particular case I,, = 0.

3. Now we carry out the summation indicated in (1.6) and (1.7). Let us note some simple properties of the denominator D (ah) :

With the help of these simple relations, we can easily regroup the terms to be summed up :

We use the following notations to write the final results :

The summation (1.7) is written as

After regrouping we get

Let us make use of(2.2) and (2.6) to express X, Y, U, V and f(a)f f ( a ) , . . . ; X, Y, U,V are functions of matrices of (V6) and (C3, and we obtainn after an easy calculation the result in its final form


where the coefficient d, g, .+I., v, f , h are functions only of the invariants of C. If

to = 1+1,+14, t+ = h$+Iph++14, t- = AB+Iph-+I4;

t+ - t- -+- x '= t -t-

2 1+614-I,, ks Y = 2-1.

and

are functions of the invariants The coefficients of (3.5) are

4. We have expressed the final result (3.6) in terms of the matrices deduced from C following fixed rules, with coefficients which are functions of the invariants of C. This result is valid in all the cases, even when for particular matrices there is no complete system of eigenvedors and we can no longer develop the resolvent matrix B as per (3.3) but the general result (3.5) always remains true. The expression (3.6) thus gives the general s~lut ion of the problem posed in (1.1).

Nevertheless it is interesting t o observe that in the particular cases I4 = 0 or I p = 0, I4 = 0, we can easily put the solution in the form of a total sum of a small number of terms, thanks to a method of iteration.

Let us briefly discuss the case I4 = 0. We have already deduced the special properties of the matrix C in (2.11) and

(2.12) : to wit


and look for a n h t i o n by putting T = U+@,.

The equation having the form U = T+8T.

we easily deduce for the equation -8U = @1+8cP8.

By repeating the operation, we successively obtain

Beacause of special properties of the operator l(2.11) - (2.1211 we easily deduce

By putting 6 =. A - E, where E is the unit operator, the equation (4 .2) is written as

(4 .4) A(@3+IEU) = G U . The original equation is

AT = U .

With det A # 0, the solution is unique and one easily sees that the solution of (4.4) is

By writing @3 = T - U+6U--88U = I.$(T- U), .

we easily have

By developing this, we have the following result :


The general solution gives us the same result by putting I4 = 0, and using the particular relations (2.11) and (2.12).

We can similarly treat the other cases I2 = 0, I4 = 0.

Post-script

If the coefficients h of

are chosen in the algebra [V]& these numbers form a new algebra [cVIz4. To solve

U = [e(EE)+e(cc)+2c(Ec)] T zz BT

is thus equivalent to finding the inverse of B as an algebraic number. The inverse is also a number in [cVl and we can easily find i t in the following way.

First, we verify the following results as per the rule of multiplication and the identities for the algebra WIG.

If = ( E ) , M = 2(EC2)+12(CC),

R = 2L+12M, X = 4(EC)2 = 2(CC)+2(EC2),

and D = 1: - 414 is the discriminant of the equations x2+12x+I, = 0 ,

then we have L(EC) = M(EC) = 0; d'oh LX = M X = RX = 0 R L = DL, R M = DM, R R = DR, M 2 = R

and XX,+212X+D = R.

In what follows we use the the following abbreviations : s = 2 + 12, to = 1 + I 2 + I4 x = 1 + 514-12, andy = 2-12.

With B = (EE) + (CC) + 2C(EC), by multiplying it with its conjugate

we have

or

B* = (EE)+(CC)-2C(EC),

BB*' = (EE+ CC)2-4C2(EC)2

BB* = ( I + ?I&!! ) ( t 0 + ~ - c 2 ~ ) to

336 S N Base : The Man and His Work

after factorization ; with ( 1 - - " ; ~ " ) ( l + ~ ] I t ,

after adding the same factor in the numerator and denominator of the right side of the equation.

By multiplying we obtain for the denominator

"

We can thus easily express the inverse by the following equation :

By eliminating the brackets, we obtain a result equivalent to the one in the main paper.

EngBtsh translation of Bull Soc Math France 88, pp. 81-88, 1955 IGauthier - Villars Publishers, Paris).

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